This printer should work in Linux (see
https://www.openprinting.org/printer...OfficeJet_5600)
HP printers have their own driver package, hplip. You need to have this installed; it's in the Mint repository so you can use synaptic to install it just like any other package.
Then you need to register the printer with cups, the Linux printing software. hplip includes a user interface called HP Setup to do this but if it doesn't work for you, you can set up your printer directly within cups. Just point your browser to localhost:631 and follow instructions. If you are lucky, your printer will be detected and registered automatically. If not, you can enter it by hand.
I apologise if you have done all this already, but your post was not very informative.
When you try to print, what happens? Do you get error messages, does the job just remain in the queue without being processed, or does the request seem to go through to completion but without anything actually being printed? The answer will determine what to try next.
