Code to bulk delete:
find /home -maxdepth 2 -type f -mtime +3 -name 'backup-*.tar.gz'
If you want to search and delete in home 2 or home 3 instead of home, adjust the code accordingly.
Example code for home 4:
find /home4 -maxdepth 2 -type f -mtime +3 -name 'backup-*.tar.gz'