Is there a mechanic or anyway to “pass” on your turn?
…just quit
There are a couple of abilities that end the turn (Invisible Woman’s green and Hood’s yellow). Other than that, no.
No – there is always a match available. If there isn’t then the board will be shuffled.
You may not like the match that’s available, particularly if it’s the only match, but that’s exactly why there’s no way to pass a turn (other than the two abilities Misguided mentions)
In older PQ games, you were able to pass your turn by making an incorrect match (which yielded a damage penalty)
Here? Not so lucky.
Looks like someone wants to full his wolverine vs lvl1 hammers 
It sucks when the match happens to include modern hawkeyes coundown tile that was just placed. I’ve had that happen twice, board shuffles, AI gets a good match.
All in the luck of the game. Sometimes I see a match coming which is going to royally screw me, but you gotta take it on the chin and man up. Let the poor AI have it’s day a few times or it will cry 8)