5610 hp printer shows insert cartridge on right when i changed to new cartridges. they are new cartridges and are inserted correctly

Try this

- Remove the cartridge that is not working. - look for a lint free cloth "coffee filter" then dump w/ water - clean the copper contacts of the cartridge - make sure that the air vent is open; if not pinch it a bit - remove all the cartridges - check if there are are inks spills - use lint free cloth to clean the part where you are have trouble with - install the inks again - run alignment and clean cartridges on the panel
if still nothing works, call HP tech support and ask for replacement of the ink.
note: they will do the same steps again for troubleshooting and documentations. the reason why this happens: - that the copper contact does not touch each other - cartridge air vent is sealed - not an orig HP ink - Print head problem, needs cleaning