The 8620 is one of the few HP printers that can use non-HP ink. You must have firmware version 1910A. Download it here if you have a different version. Any other version may not let you print if the ink is not genuine HP.
If you subscribed to Instant Ink I think you have to use Instant ink and you need to have the printer online so it can phone home. Our expert @Repairatrooper might be able to confirm this. I have no experience with instant ink.
It is always possible the cartridge is damaged or not inserted correctly or the printhead needs to be cleaned.
See if EWS can provide more information.
This explains how to use the EWS
Use the HP printer Embedded Web Server (EWS)