KK with A high flop

It depends on the villain how I would play the flop and turn. If he's loose passive I would make a thin value Cbet and for prortection and if he's a decent player I would check/call flop and go from there to keep the pot small.

As played I would call the smallish bet and her-evaluete the river. if he would bet so small agains I would make a hero call and fold if he would make it more like 40c+

Cheers,
SDK1987

I would check/call OTF and prolly same OTT.

btw, size of cbet seems fine to me

I think you played decently.
I would mix flop with some 1/4 cbets and checks OOP. You can cbet a bit more with the K♣: but I prefer check. I would play the whole range as check/call (strong and weak Axs, KQ, suited-connectors). Maybe check/raise with 9♣:8♣:. Check/fold missed crap.
Without the backdoor nutflushdraw I don't check/folding OTF sometimes and I don't think it will be too exploitable.

I would consider overbetting the turn sometimes as played or check/raise against small sizes, specially with no flush-blockers because you need more protection . This especific combo check/fold is the line because you block bluffs.
A3s, 87s are also good candidates to overbetting for instance.

Villain's turn sizing is weird btw, but I believe you can't and probably shouldn't start to speculate why because people do lots of weird stuff. It looks value heavy and this is not the top of your range, so check/fold is totally acceptable OOP.