I really don't recommend using the Disk Cleanup utility that ships with Windows. I know many people use it without issues, but some people do experience unexpected data loss due to it, including printer drivers. Deleting unnecessary files and defragging the hard drive are things I personally prefer to use third-party tools for. TuneUp Utilities, Paragon, etc.
If you reinstall the HP drivers after running Disk Cleanup, do the printer drivers stay functioning as normal until the next Cleanup run, or do you have to reinstall them daily or something?
If you insist on using Disk Cleanup, you may want to get your geek on
with this article and see if it corresponds with your problem.