Why is my printer making an awful grinding noise when printing?

Likely a worn plastic gear somewhere. These low-end tricolor HPs are cheaply built and use plastic in high-wear areas which eventually causes this problem after it gets to be older (as well as electrical issues because the copper pads where the ink sits get damaged electrically due to poor design shorting the connection). If you really want to see if it can be fixed take it apart and check which gear is worn out, then look for a donor with good gears/print mech. Otherwise, it’s better to live with the noise and let it die using $5-8 aftermarket cartridges so you're not out $40 for OEM ink on a printer with potentially a very short life left. I’d probably refill the carts myself for cheap at that point since these use dye ink and let the printer go when they stop printing right after 5-10 refills and several cleanings.

They usually run that way for a long time but given enough time they ALL fail after it starts, and the printer will be a total loss. HP knows this and uses it to push new models which use ink the 3rd party market has not remanufactured yet. It's time to replace these grinding printers when they keep jamming; the gearset is really degraded at that point.