The line to which it refers in the 'hard to read' screen shot of the file backup_helper.class.php (125) is under this comment:
// Delete all files in the curent directory return false and halt if a file cannot be removed
specifically this line:
chmod($dir_files[$i], $CFG->directorypermissions);
In Ubuntu (for example), the apache service runs under www-data user and www-data group (typically). That user/group should have read/write access to moodledata/temp/backup (the directory Moodle uses to 'build' backup files as well as restores courses).
The path indicates the system is on a shared host??? /homepages/33/d38582770/
So you might need to submit a help desk ticket with your provider to sort some of this out.
So what does the following show:
cd /pathto/moodledata/
ls -lR temp
The Coding error detected line that says it must be fixed by a programmer indicates existing temptables and suggest they need to be removed.
Does the DB user have access rights high enough to remove tables from DB for Moodle?
If you can connect to the DB from command line:
mysql> show grants for current_user;
Also, on shared systems (IF I am correct in that guess), there could be something running that helps secure a multiuser/multihosted system from other 'customers'. Sometimes, tweaks to those might be needed.
'spirit of thinkering', Ken