2NL KK OOP, cbet called, A on turn

As most of you know I have no HUD or HEM at the moment (hence playing at 2NL), so no stats.

Villain's been stationy, bought in full but has called a fair few pre and not reloaded. No hands with him personally and multi-tabling so no post flop reads.

PokerStars No-Limit Hold'em, €0.02 BB (9 handed) - PokerStars Converter Tool from FlopTurnRiver.com

MP1 (€1.04)
MP2 (€2.01)
MP3 (€0.28)
CO (€0.83)
Button (€1.95)
SB (€2.24)
BB (€2.02)
Hero (UTG) (€2.27)
UTG+1 (€0.67)

Preflop: Hero is UTG with K, K
Hero bets €0.08, 1 fold, MP1 calls €0.08, 6 folds

Flop: (€0.19) 8, J, 2 (2 players)
Hero bets €0.14, MP1 calls €0.14

Turn: (€0.47) A (2 players)
Hero?

Preflop is kinda overbet because I raised $0.6 pf the last two hands and got folded round and thought yay KK, change in raise amount might get bad calls.

Kinda want to v-bet to get one more street from his Jx etc, but wondering if so much of his range hasn't caught up that barrelling is turing my hand into a bluff. Is c/c better? If we bet, how much? What do we do if we see a river in either situation?

Bigger is fine vs these fish, could probably make it 7BB pre and overbet the flop.

Would bet/call $0.35 now. Ship rest on all but the worst rivers.

Originally Posted by Hoopy

Bigger is fine vs these fish, could probably make it 7BB pre and overbet the flop.

Would bet/call $0.35 now. Ship rest on all but the worst rivers.

This