That looks like some type of paper guide or switchback gate. I don't have a 6960 in the office I can reference, but if you take the paper tray out of the printer and set it on its back, best guess is that it goes on the back underside of the printer or in the very back of the printer if its accessible. If your having problems duplex printing, that's probably some type of switchback gate so when the printer pulls the paper back into the printer for the backside print, it can change the direction of the paper path. its looks like there 2 nipples on either end of that piece of plastic, so those probably just snap into place inside the printer. sometimes those gates have a solenoid that will flip them up or down, so you may see a little arm that looks likes its missing a friend. Non serviceable printer, so private techs don't have any service manuals to reference. Best of luck!
I am not a HP employee. I am a service tech on these forums researching topics and offering my best guess to aide in your troubleshooting.