NL10: Cbet QQ facing A on turn

[Perdix]

I'm always having trouble with my high PPs when an overcard appears on the board.

Villain is 36/9/3.3 over only 45 hands.

Poker Stars, $0.05/$0.10 NL Hold'em Cash Game, 8 Players
LeggoPoker Hand History Converter

CO: $3.40
BTN: $9
SB: $1.90
BB: $6.25
Hero (UTG): $9.85
UTG+1: $4.15
MP1: $6.70
MP2: $13.80

Pre-Flop: Q[img]/images/graemlins/heart.gif[/img] Q[img]/images/graemlins/club.gif[/img] dealt to Hero (UTG)
<font color="red">Hero raises to $0.40</font>, UTG+1 folds, MP1 calls $0.40, 5 folds

Flop: ($0.95) T[img]/images/graemlins/spade.gif[/img] 7[img]/images/graemlins/spade.gif[/img] 3[img]/images/graemlins/diamond.gif[/img] (2 Players)
<font color="red">Hero bets $0.70</font>, MP1 calls $0.70

Turn: ($2.35) A[img]/images/graemlins/diamond.gif[/img] (2 Players)
<font color="red">Hero bets $1.20</font>, MP1 calls $1.20

River: ($4.75) 7[img]/images/graemlins/club.gif[/img] (2 Players)
Hero checks, <font color="red">MP1 bets $2.30</font>, Hero ?

The Cbet is probably standard. What do I do on the turn? Give up and check or represent the ace myself (maybe betting around 1.8?)? What's my line on the river then?