Just a quick question, yes or no will work (barring the post limitations):

Can I use the effects of an Objective if it's got a focus counter on it?

VG Buzzsaw said:

Yes, in the rulebook it says abilities on a card are still usable when the card is exhausted (unless otherwise stated).

Yes, but only if the action doesn't require you to focus the objective (exhausted cards cannot have focus tokens placed on them)