KK postflop

I think he calls with better hands on the turn with Ace hearts on board and cold call on MP preflop he hasn't many flush draw combonation here. I think if he has Ax he still call here and on the river if I large bet I can throw his some combo Ax but it is risky.

I would xc on the flop to bluff catch worst hand and x/decide on the turn.

Ok and if opponent check back on the flop what's plan on the turn? Bet or check-call, check-fold?

Then you are probably ahead and can consider making a bet up to 50% of the pot.

If we can make range bet here with tight UTG range then use smaller sizing like 1/3 pot. MP range should be tight as well though so is it really spot to range cbet? My guess would be no.

KK itself is kind of hand that is first to start checking if having check range as better obviously doesn't fold and not that many weaker call. Also as we don't have heart our possibility to continue aggression after getting called decreases. Ending to lot of these turn x-f lines which in general we should avoid especially with hand that has high value in checked down scenario.

This hand is interesting.
After considering the board on the flop being not good for our OOP hero due to board favouring villians' range,( I'm not entirely sure about this) I like the fold by hero because possible flush draw aces/diamonds and possible straight draw ( turn card two of. diamonds makes the straight draw possible therefore, Bad turn card for hero.
Our pocket kings OOP now have reduced equity due to turn card and board texture.
Definitely by the action by villian MP betting the turn tells us he has connected with the board in someway and it appears villian has the range advantage here.
I don't know what hands to put Villian on ( villian called preflop) but I'm thinking likely hero is way behind by the turn and I'm folding for the most plus EV action.

@UP
There is huge problem with betting there on flop - KK doesnt achieve anything by betting flop. You are not folding better hand and dont get value by many worse. Our range also doesn't want to bet range there so check/call is only sensible option. Then on turn we also mostly check/fold but his range will be a lot wider than if we bet flop.