auto_backup_running is the name of the entry in the mdl_config_plugins table. Sometimes the backups hang and then that setting tells moodle that they are still running and won't allow them to run again.
As for the php file, the part below basically does the same as the manual change in the database. It just does it before every backup which was needed back in the 2.1 days.
backup/util/helper/backup_cron_helper.class.php is the file to be adjusted.
I placed the line
set_config('backup_auto_running', '0', 'backup');
Immediately before the line
trace("Checking automated backup status",'...');
Reading back, it looks like you have some corrupted files or database entries. It seems like maybe you could delete the old backups on that specific course as Ken suggested, then I would suggest clearing out the tables I suggested again and see if you can get it to run.