I have built new versions of my hplip-print-scan package to support HP printers up to 2022.
This is a combo package containing four drivers in PET form - 32bit bionic, 64bit bionic, 32bit slack, 64bit slack. The "bionic" versions should also work with newer Ubuntu-based Puppies. The "slack" versions work with several Slackware-based Puppies.
Extract the package, select the PET appropriate for your Puppy and left-click install it.
The driver supports print+scan for HP models up to 2019 and print-only for newer models up to 2022. It handles both USB and network connections.
I will upload the package to mediafire if someone will volunteer to test it.