Something wierd must be happening recently in Win10 because only this week, both my Win10 desktops quit working with my HP 8620 printer -- clicking the shorcut on the desktop prompted me to install a printer. And, this only worked on one desktop.
On the other, I had to remove the HP software (and that required inserting the original Printer CD!) and completely reinstall the HP software from scratch.
And, unfortunately, this last procedure was the ONLY way I was able to restore the printer desktop shortcut on the second PC.
I'm afraid that you might have to do the same -- uninstall using the CD, and reinstall from scratch.
I've recently downloaded the latest updates and still have no icon on my desktop.
I downloaded the Scan and Print Doctor, did the test scan and had the scanning window open on the computer so I could choose pdf, which was great. But still need to see ir when I scan so I can save where I want.