Thanks for the update.
It's seems that your mysql directory is using 5.6GB disk space on your server. You can remove old logs from your /var/log directory or you can move your mysql directory to other partition to make free space on /var partition
If you want to move your mysql directory to any other partition then you can try with following steps.
You can symlinking the old /var/lib/mysql to the new location
Code:
ln -s /home/var_mysql/mysql /var/lib/mysql
First make a full mysqldump file
Code:
mysqldump --all-databases | gzip > /home/alldatabases.sql.gz
Uncheck monitor in WHM > Service Manager for Mysql and save the area and stop mysql service.
Make the directory for MySQL in /home, move it and symlink it:
Code:
mkdir /home/var_mysql
mv /var/lib/mysql /home/var_mysql
chown -R mysql:mysql /home/var_mysql/mysql
ln -s /home/var_mysql/mysql /var/lib/mysql
/etc/init.d/mysql start
Re-check monitor in WHM > Service Manager for MySQL and save the area
If anything goes wrong, you have the full mysqldump backup.