Problems with legimate software percieved as malware by Apple

I've had the same issue, and it appears to be related to an ongoing problem between HP printer drivers and the latest update to Catalina. There's an discussions here: https://h30434.www3.hp.com/t5/Printing-Errors-or-Lights-Stuck-Print-Jobs/Can-t-get-around-this-Error-telling-me-to-reinstall-software/m-p/7824144/Pop_SU#M895003 and https://discussions.apple.com/thread/251948187 , amongst others.

I wouldn't change any settings, and instead wait for HP/Apple to resolve this.

I see - thank you for the clarification. Here is a tip I have found from this support article. Give it a shot, and let me know if it works:

If you're prompted to open Finder: control-click the app in Finder, choose Open from the menu, and then click Open in the dialog that appears. Enter your admin name and password to open the app.

Hey MJShifnal!

Running software that hasn’t been signed and notarized may expose your computer and personal information to malware that can harm your Mac or compromise your privacy. If you’re certain that an app you want to install is from a trustworthy source and hasn’t been tampered with, you can temporarily override your Mac security settings to open it. 

In macOS Catalina and macOS Mojave, when an app fails to install because it hasn’t been notarized or is from an unidentified developer, it will appear in System Preferences > Security & Privacy, under the General tab. Click Open Anyway to confirm your intent to open or install the app.

Hope this helps!

I can't print from any of my Macs to my network printers this AM (HP and Xerox). I can print from my iphone. I can ping and access the printer from a browser.

I think the silent updates from last night hosed printing. You can see several printer issues crop up over night in the forums (printers that works now don't, "software name" will damage your computer, network printing that won't work).

I expect more posts to follow today. I've been in IT my whole working career. I've been through the standard checks. Apple needs to fix this.

Had this pop up on my desktop last Friday. After discussions with Apple, I removed numerous programs that were uncompatible with Catalina 10.15 from legacy macs. That took care of the problem along with an annoying continuing Adobe upgrade pop up. Had to reinstall the driver for my HP 8100printer, but that was it. Good to go from there....

Are you sure it is the OS that is flagging up these files and not an anti-virus app you have installed.