AJo - Good Turn Barrel?

In my opinion (without stats) I would Cbet on the river with that gutshot. If the 7 of 9 was a club I would check/fold. It is a weakgutshot what you said, just on the line. Also depends on what kind of player villain is.
I would check/fold on the turn. I think you do not have enough equity to continue (not sure about that).

You got to be a bit more careful on these middling type boards think of villains range here vs a 3bet from mp what floats ? If we assume KK+4bet then 77-QQ TJs KQs and AK AQ likely just folds flop.Most of villains range connects with the board in some way they dont really have any really weak floats.You likely get 88 KQs to fold on the turn but not much else i don't think it's a terrible spot to barrel but likely better to check back turn and realise your hands equity.I do like cbetting though as it does allow us to realise our equity more effectively.The K or Q make much better barrels as they give you the fold equity your after where as the 4 doesn't really change much for villains range.