diff --git a/src/Interpreters/Session.cpp b/src/Interpreters/Session.cpp
index f0bb339e278..64f7b4fc934 100644
--- a/src/Interpreters/Session.cpp
+++ b/src/Interpreters/Session.cpp
@@ -107,7 +107,7 @@ public:
         if (it == sessions.end())
         {
             if (throw_if_not_found)
-                throw Exception(ErrorCodes::SESSION_NOT_FOUND, "Session not found.");
+                throw Exception(ErrorCodes::SESSION_NOT_FOUND, "Session {} not found", session_id);
 
             /// Create a new session from current context.
             auto context = Context::createCopy(global_context);
@@ -129,7 +129,7 @@ public:
             LOG_TEST(log, "Reuse session from storage with session_id: {}, user_id: {}", key.second, key.first);
 
             if (!session.unique())
-                throw Exception(ErrorCodes::SESSION_IS_LOCKED, "Session is locked by a concurrent client.");
+                throw Exception(ErrorCodes::SESSION_IS_LOCKED, "Session {} is locked by a concurrent client", session_id);
             return {session, false};
         }
     }
diff --git a/src/Storages/RocksDB/StorageEmbeddedRocksDB.cpp b/src/Storages/RocksDB/StorageEmbeddedRocksDB.cpp
index 9ff0c152399..20839982ec3 100644
--- a/src/Storages/RocksDB/StorageEmbeddedRocksDB.cpp
+++ b/src/Storages/RocksDB/StorageEmbeddedRocksDB.cpp
@@ -276,7 +276,7 @@ void StorageEmbeddedRocksDB::mutate(const MutationCommands & commands, ContextPt
 
     assert(commands.front().type == MutationCommand::Type::UPDATE);
     if (commands.front().column_to_update_expression.contains(primary_key))
-        throw Exception(ErrorCodes::BAD_ARGUMENTS, "Primary key cannot be updated");
+        throw Exception(ErrorCodes::BAD_ARGUMENTS, "Primary key cannot be updated (cannot update column {})", primary_key);
 
     auto interpreter = std::make_unique<MutationsInterpreter>(
         storage_ptr, metadata_snapshot, commands, context_, /*can_execute_*/ true, /*return_all_columns*/ true);
diff --git a/src/Storages/StorageKeeperMap.cpp b/src/Storages/StorageKeeperMap.cpp
index f570f132463..5f7726e11bf 100644
--- a/src/Storages/StorageKeeperMap.cpp
+++ b/src/Storages/StorageKeeperMap.cpp
@@ -827,7 +827,7 @@ void StorageKeeperMap::mutate(const MutationCommands & commands, ContextPtr loca
 
     assert(commands.front().type == MutationCommand::Type::UPDATE);
     if (commands.front().column_to_update_expression.contains(primary_key))
-        throw Exception(ErrorCodes::BAD_ARGUMENTS, "Primary key cannot be updated");
+        throw Exception(ErrorCodes::BAD_ARGUMENTS, "Primary key cannot be updated (cannot update column {})", primary_key);
 
     auto interpreter = std::make_unique<MutationsInterpreter>(
         storage_ptr, metadata_snapshot, commands, local_context, /*can_execute_*/ true, /*return_all_columns*/ true);
diff --git a/tests/clickhouse-test b/tests/clickhouse-test
index 9067a8142bc..e0d51638593 100755
--- a/tests/clickhouse-test
+++ b/tests/clickhouse-test
@@ -2076,7 +2076,10 @@ def reportLogStats(args):
               'AST is too deep. Maximum: {}', 'Array sizes are too large: {}', 'Unable to connect to HDFS: {}',
               'Shutdown is called for table', 'File is not inside {}',
               'Table {} doesn''t exist', 'Database {} doesn''t exist', 'Table {}.{} doesn''t exist',
-              'File {} doesn''t exist', 'No such attribute ''{}''', 'User name ''{}'' is reserved'
+              'File {} doesn''t exist', 'No such attribute ''{}''', 'User name ''{}'' is reserved',
+              'Could not find table: {}', 'Detached part "{}" not found', 'Unknown data type family: {}',
+              'Unknown input format {}', 'Cannot UPDATE key column {}', 'Substitution {} is not set',
+              'Cannot OPTIMIZE table: {}', 'User name is empty', 'Table name is empty', 'AST is too big. Maximum: {}'
         ) AS known_short_messages
         SELECT count() AS c, message_format_string, substr(any(message), 1, 120)
         FROM system.text_log
@@ -2084,7 +2087,7 @@ def reportLogStats(args):
             AND (length(message_format_string) < 16
                 OR (length(message_format_string) < 30 AND message ilike '%DB::Exception%'))
             AND message_format_string NOT IN known_short_messages
-        GROUP BY message_format_string ORDER BY c DESC LIMIT 30 FORMAT TSVWithNamesAndTypes
+        GROUP BY message_format_string ORDER BY c DESC LIMIT 50 FORMAT TSVWithNamesAndTypes
     """
     value = clickhouse_execute(args, query).decode(errors="replace")
     print("\nTop short messages:\n")
diff --git a/tests/queries/0_stateless/00002_log_and_exception_messages_formatting.reference b/tests/queries/0_stateless/00002_log_and_exception_messages_formatting.reference
index fddfbd49de3..60ac6e30c59 100644
--- a/tests/queries/0_stateless/00002_log_and_exception_messages_formatting.reference
+++ b/tests/queries/0_stateless/00002_log_and_exception_messages_formatting.reference
@@ -1,8 +1,8 @@
 runtime messages	0.001
 runtime exceptions	0.05
-messages shorter than 10	10
-messages shorter than 16	40
-exceptions shorter than 30	120
+messages shorter than 10	0
+messages shorter than 16	2
+exceptions shorter than 30	40
 noisy messages	0.3
 noisy Trace messages	0.16
 noisy Debug messages	0.09
diff --git a/tests/queries/0_stateless/00002_log_and_exception_messages_formatting.sql b/tests/queries/0_stateless/00002_log_and_exception_messages_formatting.sql
index 71f41c7a9d2..b9269b66ff7 100644
--- a/tests/queries/0_stateless/00002_log_and_exception_messages_formatting.sql
+++ b/tests/queries/0_stateless/00002_log_and_exception_messages_formatting.sql
@@ -14,14 +14,43 @@ select 'runtime messages', max2(coalesce(sum(length(message_format_string) = 0)
 -- Check the same for exceptions. The value was 0.03
 select 'runtime exceptions', max2(coalesce(sum(length(message_format_string) = 0) / countOrNull(), 0), 0.05) from logs where message like '%DB::Exception%';
 
+-- FIXME some of the following messages are not informative and it has to be fixed
+create temporary table known_short_messages (s String) as select * from (select
+['', '({}) Keys: {}', '({}) {}', 'Aggregating', 'Became leader', 'Cleaning queue', 'Creating set.',
+'Cyclic aliases', 'Detaching {}', 'Executing {}', 'Fire events: {}', 'Found part {}', 'Loaded queue',
+'No sharding key', 'No tables', 'Query: {}', 'Removed', 'Removed part {}', 'Removing parts.',
+'Request URI: {}', 'Sending part {}', 'Sent handshake', 'Starting {}', 'Will mimic {}', 'Writing to {}',
+'dropIfEmpty', 'loadAll {}', '{} ({}:{})', '{} -> {}', '{} {}', '{}: {}', 'Query was cancelled',
+'Table {} already exists.', '{}%', 'Cancelled merging parts', 'All replicas are lost',
+'Cancelled mutating parts', 'Read object: {}', 'New segment: {}', 'Unknown geometry type {}',
+'Table {} is not replicated', '{} {}.{} already exists', 'Attempt to read after eof',
+'Replica {} already exists', 'Convert overflow', 'key must be a tuple', 'Division by zero',
+'No part {} in committed state', 'Files set to {}', 'Bytes set to {}', 'Sharding key {} is not used',
+'Cannot parse datetime', 'Bad get: has {}, requested {}', 'There is no {} in {}', 'Numeric overflow',
+'Polygon is not valid: {}', 'Decimal math overflow', '{} only accepts maps', 'Dictionary ({}) not found',
+'Unknown format {}', 'Invalid IPv4 value', 'Invalid IPv6 value', 'Unknown setting {}',
+'Unknown table function {}', 'Database {} already exists.', 'Table {} doesn''t exist',
+'Invalid credentials', 'Part {} already exists', 'Invalid mode: {}', 'Log pulling is cancelled',
+'JOIN {} cannot get JOIN keys', 'Unknown function {}{}', 'Cannot parse IPv6 {}',
+'Not found address of host: {}', '{} must contain a tuple', 'Unknown codec family: {}',
+'Expected const String column', 'Invalid partition format: {}', 'Cannot parse IPv4 {}',
+'AST is too deep. Maximum: {}', 'Array sizes are too large: {}', 'Unable to connect to HDFS: {}',
+'Shutdown is called for table', 'File is not inside {}',
+'Table {} doesn''t exist', 'Database {} doesn''t exist', 'Table {}.{} doesn''t exist',
+'File {} doesn''t exist', 'No such attribute ''{}''', 'User name ''{}'' is reserved',
+'Could not find table: {}', 'Detached part "{}" not found', 'Unknown data type family: {}',
+'Unknown input format {}', 'Cannot UPDATE key column {}', 'Substitution {} is not set',
+'Cannot OPTIMIZE table: {}', 'User name is empty', 'Table name is empty', 'AST is too big. Maximum: {}'
+] as arr) array join arr;
+
 -- Check that we don't have too many short meaningless message patterns.
-select 'messages shorter than 10', max2(countDistinctOrDefault(message_format_string), 10) from logs where length(message_format_string) < 10;
+select 'messages shorter than 10', max2(countDistinctOrDefault(message_format_string), 0) from logs where length(message_format_string) < 10 and message_format_string not in known_short_messages;
 
 -- Same as above. Feel free to update the threshold or remove this query if really necessary
-select 'messages shorter than 16', max2(countDistinctOrDefault(message_format_string), 40) from logs where length(message_format_string) < 16;
+select 'messages shorter than 16', max2(countDistinctOrDefault(message_format_string), 2) from logs where length(message_format_string) < 16 and message_format_string not in known_short_messages;
 
 -- Same as above, but exceptions must be more informative. Feel free to update the threshold or remove this query if really necessary
-select 'exceptions shorter than 30', max2(countDistinctOrDefault(message_format_string), 120) from logs where length(message_format_string) < 30 and message ilike '%DB::Exception%';
+select 'exceptions shorter than 30', max2(countDistinctOrDefault(message_format_string), 40) from logs where length(message_format_string) < 30 and message ilike '%DB::Exception%' and message_format_string not in known_short_messages;
 
 
 -- Avoid too noisy messages: top 1 message frequency must be less than 30%. We should reduce the threshold
diff --git a/tests/queries/0_stateless/00463_long_sessions_in_http_interface.sh b/tests/queries/0_stateless/00463_long_sessions_in_http_interface.sh
index 89da84a5bdd..35e75c9ec4e 100755
--- a/tests/queries/0_stateless/00463_long_sessions_in_http_interface.sh
+++ b/tests/queries/0_stateless/00463_long_sessions_in_http_interface.sh
@@ -8,7 +8,7 @@ CURDIR=$(cd "$(dirname "${BASH_SOURCE[0]}")" && pwd)
 
 
 echo "Using non-existent session with the 'session_check' flag will throw exception:"
-${CLICKHOUSE_CURL} -sS "${CLICKHOUSE_URL}&session_id=nonexistent&session_check=1" --data-binary "SELECT 1" | grep -c -F 'Session not found'
+${CLICKHOUSE_CURL} -sS "${CLICKHOUSE_URL}&session_id=nonexistent&session_check=1" --data-binary "SELECT 1" | grep -c -F 'SESSION_NOT_FOUND'
 
 echo "Using non-existent session without the 'session_check' flag will create a new session:"
 ${CLICKHOUSE_CURL} -sS "${CLICKHOUSE_URL}&session_id=${CLICKHOUSE_DATABASE}_1" --data-binary "SELECT 1"
@@ -30,7 +30,7 @@ ${CLICKHOUSE_CLIENT} --multiquery --query "DROP USER IF EXISTS test_00463; CREAT
 ${CLICKHOUSE_CURL} -sS -X POST "${CLICKHOUSE_URL}&session_id=${CLICKHOUSE_DATABASE}_6&session_timeout=600" --data-binary "CREATE TEMPORARY TABLE t (s String)"
 ${CLICKHOUSE_CURL} -sS -X POST "${CLICKHOUSE_URL}&session_id=${CLICKHOUSE_DATABASE}_6" --data-binary "INSERT INTO t VALUES ('Hello')"
 
-${CLICKHOUSE_CURL} -sS "${CLICKHOUSE_URL}&user=test_00463&session_id=${CLICKHOUSE_DATABASE}_6&session_check=1" --data-binary "SELECT 1" | grep -c -F 'Session not found'
+${CLICKHOUSE_CURL} -sS "${CLICKHOUSE_URL}&user=test_00463&session_id=${CLICKHOUSE_DATABASE}_6&session_check=1" --data-binary "SELECT 1" | grep -c -F 'SESSION_NOT_FOUND'
 ${CLICKHOUSE_CURL} -sS -X POST "${CLICKHOUSE_URL}&user=test_00463&session_id=${CLICKHOUSE_DATABASE}_6&session_timeout=600" --data-binary "CREATE TEMPORARY TABLE t (s String)"
 ${CLICKHOUSE_CURL} -sS -X POST "${CLICKHOUSE_URL}&user=test_00463&session_id=${CLICKHOUSE_DATABASE}_6" --data-binary "INSERT INTO t VALUES ('World')"
 
@@ -53,7 +53,7 @@ do
         ${CLICKHOUSE_CURL} -sS "${CLICKHOUSE_URL}&session_id=${CLICKHOUSE_DATABASE}_7&session_timeout=1" --data-binary "SELECT 1"
         ${CLICKHOUSE_CURL} -sS "${CLICKHOUSE_URL}&session_id=${CLICKHOUSE_DATABASE}_7&session_check=1" --data-binary "SELECT 1"
         sleep 3
-        ${CLICKHOUSE_CURL} -sS "${CLICKHOUSE_URL}&session_id=${CLICKHOUSE_DATABASE}_7&session_check=1" --data-binary "SELECT 1" | grep -c -F 'Session not found'
+        ${CLICKHOUSE_CURL} -sS "${CLICKHOUSE_URL}&session_id=${CLICKHOUSE_DATABASE}_7&session_check=1" --data-binary "SELECT 1" | grep -c -F 'SESSION_NOT_FOUND'
     ) | tr -d '\n' | grep -F '111' && break || sleep 1
 done