I solved this and the inability to print by following these steps exactly:
Go to 123.hp.com
Do NOT put in your printer type, instead use the "My device is not listed" link. This takes you to a more generalized tool to download.
Click the Download button. The application "HP Easy Start" version 2.9.3 will download.
Open this app... do NOT go beyond the first screen. Instead, look under the HP Easy Start menu next to the Apple menu, and select and choose "Uninstall HP Software..."
This launches the Uninstaller. Click Continue. Then it takes quite a while to remove all HP files.
When it's done, there is a button to relaunch HP Easy Start, called "Set Up New Device". Click that.
This time "Agree" and hit the Continue button. Make sure of the listed things on the next screen and click Continue.
Select your printer (mine is a printer on the network, it just shows up by its name, e.g. "Anubis") and click Continue.
Click "Install" in the next panel. The 'essential' software is downloaded, and permission must be given to proceed with installing.
Installation seems to take a long time. When it is done click Continue.
At this point you can set up Status and Alerts. I don't know what happens if you do, I just skipped that step by clicking Continue.
In the next panel, click "Add Device..." button. That doesn't take long. Then click "Print Test Page".
If you are lucky like me that page prints. Click Continue. Next is to set up online access things. I didn't really want that, so when I clicked the "Continue Online..." button I did NOT click the "Register My Printer" button in the web page that opened. You might want to.
I confirmed that printing from an app actually works, and then restarting the computer did NOT result in the "damage" warning.
YAY.
Worked on a Catalina iMac and a Mohave MacBook Air.
