If I have a resource that has say two focus tokens on it and I refresh at the next turn, even though I have one token left on it, do I still get the benefit of the objective even though it has a focus token on it? (able to draw an extra card at the start of my turn, or putting shields on one of my cards)?

Thanks

Unless the objective's text requires you to focus it (see the black squadron objective) you can use it's text if triggered.

MasterJediAdam is right. This can be found on pg 27 (emphasis mine):

"Exhausted
A card that is exhausted (i.e. has one or more focus tokens on it) cannot be focused to pay for card effects, cannot focus to strike, and cannot generate resources. However, other card abilities of an exhausted card may still be triggered, and its constant effects (if any) are still active, as long as such abilities do not require the card to focus, to be ready, or are not otherwise prohibited by the card text. "