This is a link to the HP linux driver (HPIJS) that uses CUPS. It seems this driver works with your printer, my 3650DJ, and 598 other printers under CUPS.
I don't know enough to compile it on this machine but I did post a request for a DotPup a while back. Maybe folks are not wanting to come out with new DotPups because Puppy 2.14 will use the new system so they are waiting til 2.14 comes out?
It seems if someone smarter than me could compile this driver for Puppy 2.13 then it would be a plus for us and a lot of other HP owners out there. I have my 2.13 so tweeked out that I won't update until maybe 2.20. I love Puppy and it is working perfectly for me but my printer that I don't really use doesn't work.
I hope this points you in the right direction.
I just read this:
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