Hello,
It may be a good idea to reset the printing system. Does the operating system offer a suitable driver to use?
Reset the Mac printing system to solve a problem - Apple Support
Add a printer on Mac - Apple Support
There is an HP Easy Admin tool for Mac that allows administrators to download drivers without a connected printer. If you need drivers/software from HP, and there is a problem with HP Easy Start or HP Smart, you may want to try this tool.
https://support.hp.com/us-en/document/c06164609
If necessary, you could perhaps try the Dec 11, 2020 HP Printer Drivers v5.1 for macOS (https://support.apple.com/kb/DL1888?viewlocale=en_US&locale=en_US) package from Apple instead. This download has been known to work with a number of models even under Big Sur. Would something like the HP LaserJet Professional P1100(w) version 6.9 driver mentioned in the Printer and scanner drivers for Mac - Apple Support article work?
If USB does not work, try wireless (or vice versa).
The HP LaserJet P1102w does not appear to be listed as a supported AirPrint model in the latest (September 30, 2021) About AirPrint - Apple Support. However, the printer is in fact mentioned in an old (February 13, 2012) iOS: AirPrint 101 HT4356 article. It cannot be ruled out that the model in question worked with an older AirPrint version. For a test with AirPrint, the printer and the computer must be connected to the same network (Wi-Fi router).
Before testing any drivers, make sure that you have backup copies of all important files (documents, photos, et cetera).