$.50 preflop for value.
As played preflop, bet at least .50 on the flop and probably more like .60-.65 because a
lot of your opponent's limp/call range is going to call at least one bet on this board.
In general, betting the turn 2/3-3/4 pot (if not full pot) is good. If he peeled the flop with broadways, he's probably folding regardless, so the plan should be to maximize value vs pair/draw hands.
The river completes everything under the sun either with a straight, two pair, or a flush. The only worse hand that
might call a bet are TP-type hands and overpairs and frankly, compared to the amount of 2pair hands, straights, flushes, sets, etc there just aren't enough worse hands that call a bet. Since betting is out, that leaves us with either check-folding or check-calling. Most of the time, when a player bets this kind of board (especially when checked to) they either have air, or something VERY strong because of how scary the board is. If you check-call in a spot like this, you'd be bluff-catching. The problem with that in this specific hand is that
everything got there. He really can't have bluffs after calling two streets when this river hits so the best play is to check-fold. Overall, the hand should play out something like this.
Hero (BB): $10.45 (104.5 bb)
MP1: $11.07 (110.7 bb)
MP2: $9.45 (94.5 bb)
MP3: $11.76 (117.6 bb)
SB: $3.64 (36.4 bb)
BTN: $10.77 (107.7 bb)
CO: $3.04 (30.4 bb)
Preflop: Hero is BB with A
A
MP1 folds, MP2 calls $0.10, 4 folds, Hero raises to $0.50, MP2 calls $0.40
Flop: ($1.05) 4
7
T:diamond (2 players)
Hero bets $0.90, MP2 calls $0.90
Turn: ($2.85) 8
(2 players)
Hero bets $2.20, MP2 calls $2.20
River: ($7.25) 6 (2 players)
Hero checks, MP2 bets $5.85 all-in, Hero folds.
This way, we build a much bigger pot for when he has a draw that doesn't get there on the river (note how much bigger this river pot is than the one in your hand). If he does get there, he wins an additional 80 cents...but he had to pay an additional $2.40 (ish) to get there.
Cliff notes: bet big for value and don't bluff-catch when all the draws come in.
As an aside, let's say the board was like this
Flop: ($0.65) 4
7
T
(2 players)
Hero bets $0.31, MP2 calls $0.31
Turn: ($1.27) Q
(2 players)
Hero bets $0.60, MP2 calls $0.60
River: ($2.47) 2
(2 players)
Now the line of
Hero checks, MP2 bets $1.60, Hero calls $1.60
makes a lot more sense (especially if the opponent is aggressive) since all the draws missed.