The mysqld: file './binlog.index' not found (os errno 13 - permission denied) error usually indicates that the MySQL server is unable to access the binlog.index file,
which is used to store information about the binary log files that the server uses to keep track of changes to the database. This error can be caused by a number of factors, including:
Permission issues: The mysqld process may not have sufficient permissions to access the binlog.index file.
Make sure that the MySQL user has the necessary permissions to read and write to this file.
Corrupt or missing file: The binlog.index file may be corrupt or missing. If the file is missing, you may need to restore it from a backup.
If the file is corrupt, you may need to delete it and let MySQL recreate it.
Incorrect file path: The MySQL configuration file (my.cnf) may specify an incorrect path to the binlog.index file.
Make sure that the path specified in the configuration file is correct.
Disk space issues: The server may be running out of disk space, which could prevent it from creating new binary log files.
Make sure that there is enough free space on the disk to allow MySQL to operate properly.
I hope this helps! Let me know if you have any other questions or need further assistance.