Your turn bet is fine. His min raise is a little strange. Has he been making this play in other hands? and with what? It kind of looks like he took a card off with A-Q-x-x and wants to see how much you like your hand. It could have even been A-Q-Q-6 or A-Q-Q-3. Anyways, even if he has a set, which you can't fully discount, you are certainly priced in to hit your flush.

I'm betting 1/3 to 1/2 pot (whatever you think gets a call) with a non-pairing club. If a Ace, 6 or 3 hits, I'm check/calling. Unless he is a complete maniac, I'm probably check/folding most other cards.