Thanks guys.
I opened the printer and took it apart as far as possible. As I stated before, the paper is not lifted high enough, which is part of the problem. As it turns out, the long spring that pulls up the paper-tray mechanism (near the white nylon cogwheels on the left) has lost much of its elasticity already.
This is a crap-printer, made with cheap parts in a cheap country (Hungary, no offense meant to the Hungarians though).
I shortened that spring by about 1 cm and put it back in. It now lifts the paper again to where it is supposed to be, BUT...
Somehow the timing seems to have got off. It pulls in a sheet, but this then gets caught in the 3 black plastic levers that are standing up in front of the roller, when looking in from the front.
The paper screws up, gets rolled backwards a little and then the printer starts printing, and of course ends up jamming.
The levers on each side, that should lift the paper when it comes out to dry, don't work correctly either.
This printer is only 8 months old, and is the second same one that my friend got.
Reading up on the internet reveals that the HP Deskjet 940c is called an "Ink-guzzler", breaks down after only a short time, has only 90 days guarantee and is a piece of shite, as we say here in Ireland.
Anyway, my friend is a well-known sculptor and designer, and seems to have some clout. He called up HP, asked to speak to the manager, got connected, chewed him out for the rotten products HP make, threatened to go public on the web with his bad HP-experiences, unless HP did something to change his mind.
The result is, that today or tomorrow a brand-new HP printer of a more advanced/modern type is to be delivered, completely free of charge and with compliments of HP.
Not bad for one phonecall!
In the meantime I have this heap sitting here on my workbench. If anyone can help me with adjusting the timing, I'd be grateful. I will donate this printer to charity when it is up and running again.
Thanks in advance for all your help.