$5 NLHE Full Ring: $5 NLHE Full Ring: Would a raise after the flop have saved me some cash?

$5 NL HE Full Ring: $5 NL HE Full Ring: Would a raise after the flop have saved me some cash?

pokerstars No-Limit Hold'em, $0.02 BB (8 handed) - Poker-Stars Converter Tool from FlopTurnRiver.com

MP1 ($1.38)
MP2 ($0.99)
CO ($5.25)
Hero (Button) ($2.02)
SB ($1.56)
BB ($0.39)
UTG ($2.75)
UTG+1 ($1.77)

Preflop: Hero is Button with 10 , 10
3 folds, MP2 calls $0.02, 1 fold, Hero bets $0.10, 2 folds, MP2 calls $0.08

Flop: ($0.23) K , 4 , 2 (2 players)
MP2 bets $0.06, Hero calls $0.06

Turn: ($0.35) 9 (2 players)
MP2 bets $0.12, Hero calls $0.12

River: ($0.59) A (2 players)
MP2 bets $0.12, Hero folds

Total pot: $0.59 | Rake: $0.02

Results below:
MP2 didn't show
Outcome: MP2 won $0.57

I don't know if it would have saved you some cash but it would give you an idea of what your opponent may have.

Yes, I guess if I got a re-raise or a call and he bet on the next street I can assume he has me. If he checks the next street I can take the free card.

I get to see the same cards for the same price but get to keep the initiative.

Flop is an easy raise then if he calls just try and keep the pot small