Screen says "Print cartridge problem" HP Photosmart C4780

This is a common issue with semi-modern integrated head (tricolor/black) HPs with the small tricolor cart. It has been a known issue for at least 10 years when HP reduced the contract copper, and the printers had less tolerance for contact failure.The HP 6X machines were some of the first to make it more commonplace. The problem is there's no fix for it short of replacing the entire print mechanism if it has permanent damage :-(. The issue is that the copper reduction over time has made the printers more problematic when it happens.

Older machines like the ones that used the HP 26/27 and 56/57 carts are less problematic, but it still (sometimes) occurs.

First, I'm not going to pretend you're a criminal for using aftermarket carts and not using genuine HP supplies—the printer is 10+ years old, and the cost of OEM was always as much as the machine. You come out ahead by using third-party ink from the start and replacing it if the ink damages it. That said, refilling an OEM core is usually cheaper until the cartridge is worn and needs to be replaced after 4-5 refills. Good reman ink is also an option. The issue is it's the Amazon junk cart that never work which cause issues. If you have bad reman ink installed, get a set from a known good brand like LD Products or Onn.

As much as HP disgusts me today due to the HP DS firmware, if people are still willing to buy them, the good thing is HP doesn't exactly retaliate against 3rd paty ink manufacturers so that stores can cut the wait and put it on the shelf when there's a strong supply. Canon is the same as the old HPs without the DRM - so that's my go-to for people who buy cheap printers and DIY the refill procedure, or buy remanufactured from day one. I don't recommend HP, but if someone is going to do it, I advise them NOT to ENABLE HP+ (this is in general) and to buy 3rd party ink if the printer was cheap to purchase initially. Once you pull a cartridge from an HP+ machine it may be blacklisted - I don't know and frankly put HP on my blacklist after that "feature" was announced.

If it's a good brand ink cartridge or OEM, look at the printer's contacts for any ink or burn marks. Use an eyeglass wipe or IPA and a napkin if you see ink. You don't need to be fancy and use a lint-free cloth or microfiber on these; they're designed to work, assuming dust will get on the contacts over time. If that fixes it, it came from the waste ink service station. You can potentially take it apart and clean the service station up. The issue is that they design them with such limited capacity, and you will do it so often. It's generally more time efficient to clean the contacts with alcohol and a napkin with every cartridge change to avoid problems. If the pads are burned on the printer end, the printer is effectively junk as it will never work right again once you get this message.