Hi - for the record, I managed to solve this for my printer:
I purchased a replacement printhead off Amazon - one from a random Chinese supplier (rather than an official HP part), which came with no setup cartridges.
Once I had fitted the replacement printhead, I fitted a set of nearly-new HP cartridges (which I had tried using in the printer since it started complaining).
I turned the printer off and on - but then got the same same printer/ink flow error... :-(
So I turned off again, and then powered off at the wall socket for 30 seconds.
I then powered on the wall socket, waited 5 seconds, and press-held the power button for 5 seconds - the display will briefly flash then go black.
Sometimes I didn't get the timing quite right - and the printer turned on as normal. In this case, I did the full power off, power off at wall for 30 seconds etc
Then press Home-Return-Home-Home buttons at about 1 second intervals; prior to doing this, I noted where the buttons are by shining a light at an oblique angle at the panel - they won't be lit, so you can't see where they are otherwise.
The service menu then came up. Using the four lit buttons round the edge of the display, choose MENU then NEXT.
The display showed SEMI RESET - so I chose OK.
Following this, the printer was then restored to working order! :-)