I had the exact same question and I figured it all out while I was at work. Here it is!
Okay a flush draw is actually 1 in 3, I don't go by that type of ratio (4:1). I usually just say the chances of making a flush draw by the river is 2 to 1.
Anyways before your opponent bets the pot is $0.13 correct? And after his bet it is now $0.23 and you must call $0.10 into a pot of $0.23.
The chances of you hitting those 9 (clean) outs on the turn is 18%. So mathematically speaking your opponent must bet $0.03 on the flop for it to be a breakeven call. $0.16 *0.18 = $0.0288
IDEAL POT:18% (1 in 5.55) = 5.55 X $0.0288
= $0.15984 (pot)
YOUR POT: $0.23
Since you called a $0.10 bet instead of $0.03, you must make up -$0.07 (you paid $0.07 more). Now the pot is $0.33 and he bets $0.20 into it.
Again if you called this bet you'd need to make up another -$0.15 on the river.
IDEAL POT: 20% (1 in 5) = 5 x $0.048
= $0.24 (pot)
YOUR POT: $0.53
$0.07 + 0.15 = $0.22
In total you gotta squeeze another $0.22 - $0.23 out of your opponent for it to be breakeven overall.
What I am trying to say is this, ideally in your situation you would've wanted your opponent to bet $0.03 instead of $0.10 on the flop to see the turn card. And you would've wanted your opponent to bet $0.05 into a now pot of $0.19 to see the river.
But because you paid $0.07 more to see the turn card and $0.15 more to see the river card, you must now make up $0.22 - $0.23 to be breakeven.
EDIT: This is the conclusion that I came to. If I am wrong in any way, please feel free to correct me because I am always willing to learn.
FYI: I did 20% on river because that's the percent I got from imputing a flush draw on cardplayer's poker odds calculator.
