Unfortunately, this appears to be an issue with HP printer drivers and/or Catalina 10.15.7 - I'm one of many people with the same issue. Lots of us are reporting the same symptoms on the HP Support forums, but at the time of writing there's no fix. Based on experiences so far, HP are ignoring the issue, hanging up on phone calls, or claiming that you'll need to buy a new printer.
All indications at the moment are that there's some kind of issue with the driver signing. Using AirPrint may be a workaround.
I'll update this answer if anything substantial changes.
UPDATE: The issue has now been resolved. We can assume that Apple and/or HP has reinstated the code signing certificate, or added the signature to a whitelist, or some other behind-the-scenes magic. No user intervention was required for me - no reinstallation of drivers or anything else, the printer that was failing to start a few days ago now works without any problems.