HI Vigster:
Welcome to the forum!
Thank you for your query. I will do my best to assist you!
The HP Tools drive is not intended for you to save you backup on, It is a very small partition that stores all your original software that came with the system. and the space is very limited, hence the low disk space error. For your backups I would recommend that you either burn or save them to a CD/DVD, a usb drive or an external drive. If you have an external drive you can still do the auto backup and point it to the external. There is also the option of using a third party software to create a new partition to save your backup to, without harming the original c drive. Please be aware if you store your backup on the computer, if the hard drive should fail for any reason, you would likely not be able to access it with doing data recovery. You can opt to do manual backups.
Storing your backup on the HP Tools drive will not harm your computer, but you will soon run into the same issue.