REAL ANSWER.
As in most HP printers, the printer heads are on a Carriage tray into which the ink Cartridges click into. The ONLY REAL way to clear the problem is take it to a local printer / PC tech service and ask them to remove AND CLEAN the Carriage Tray in a soft solvent to clear the dried residue from the actual heads themselves. This will involve unscrewing a few panels to enable the Carriage tray to be removed. Not difficult. The cartridges themselves are NOT the problem in this instance. This will happen if the printer has been idle for some time. Once gummed up pouring more printer ink (head cleaning) into an already blocked and dried up head system will not work as many have already experienced. Also a good time to clean the ADF feeding, drive and pickup rollers, coder stripe and panels. The Printer is fine - resist the urge to chuck it. I have done this and HP 4625 works well once more.