[snakes1100]

The error indicates that the max_allowed_packet setting in MySQL is to low, it needs to be increased, i would ask you host or if your on a dedicated server, use root access and increase the limit.

Dumping the databse should be done with a shell using mysqldump, you shouldn't depend on the backup from the admincp, i wouldn't trust a backup from there, phpmyadmin is also another source to backup your database, but again, mysqldump is the most reliable.