2nd hand bike, a fair amount of play at the rear linkage of the shock. Rides fine though.

Not really knowing much about the names of the stuff there, is it just a DU bush (is that a hoopla hoop (as in the crisps) bit of plastic)

Or will the aluminium spacer bits either side and into the shock eye need renewing, in your opinions obviously. Anyway, what are these bits called?

The hula hoo is the DY bushing, it’s designed to wear and save the shock eyelets. the other parts are known as mounting hardware and a few other names (top hats etc.) they can wear but it’s not as common.
This page should sort you out, there is a special tool for removing and replacing the bush. Shock eyelets can also be different sizes do you need to right one for the shock (by brand) and the tool only does one size.