I'm assuming you have completely uninstalled the printing software and re-downloaded the latest package for your printer from HP and it still does that? If not, do so and it will likely fix that issue.
If you have completed the above, I would uninstall the software and driver, run CCleaner on the registry side, manually delete the HP Printer folder in Program Files, and look through the registry for any other entries missed by CCleaner. Then reinstall from a fresh download.
If all of that doesn't work, you can just install the driver instead of the whole software package from HP and then it wont have anything that it would be trying to install.
Learning - A never ending quest for knowledge usually attained by being thrown in a situation and told to fix it NOW.