The PSC 1510 _appears_ to be supported with the latest update for HP printers. That is, it is on the list of other supported HP printers, from the Apple support page for those printers that are supported by Apple's printer software updates.
One caveat is that your PSC 1510 is an older model, 10 years, and _some_ older printers may show support, yet might not actually work reliably.
The main idea is that the original software will very likely not work, and I suggest that you should not even try to install the original HP software on your new Mac mini. When the printer is first connected and turned on, then software update (in the App Store/Updates tab) should offer the HP printer update. Test your printer after the new software installs. It may work (as some older printers will). If it does not work to your satisfaction, then I expect it is a software issue that neither HP, nor Apple will have a solution, other than replacing your printer with a newer model. HP sells printers now that will easily allow you to print from your smart phone directly (among other connection features that did not exist 10 years ago
). Your HP PSC 1510 would not even suspect that could happen.
I have a couple of the HP Envy series printers, which work wonderfully, even with one-time connections, wirelessly from visitor's laptops. I have not experienced printers that worked that simply.
Anyway, you should test the old printer on your new Mac. If your PSC 1510 decides to not work with the new Mac (although it may be just fine ... ), then you can go with the suggestion that it is time to move on to a newer printer.