Fun little mini-game
Very simple controls and fun short gameplay! I especially like the emphasis on accuracy and speed. It's an all or nothing deal, you either punch only animals or you're toast. This is one of those games that's easy to do well but much much much easier to majorly mess up. It's like a non-rhythm-centric Guitar Hero meets Minesweeper - all accuracy and no forgiveness for error. This game sets a high bar for entry into the leaderboard.
For the most part this game is fun and very entertaining. The difficulty is perfect for a game of this kind. I also like how you chose to forgo music since it could be really distracting in a game like this. If you included music I'll probably find myself punching animals to the beat.
While I do like the game as it is, I have some suggestions that I'm not sure if you'll want to take into consideration:
1. In TimeAttack mode, i tested when the timer started by waiting 30 seconds before commencing my punching. After 30 seconds I started punching at a pace that was relatively slow, slow enough to punch accurately but not so slow that it would have taken me more than 20 seconds to finish. Surely enough the counter registered my time as being 43.78 seconds. I do believe my real time in that run was 13.78 seconds not counting the 30 second lag phase.
Now, my suggestion is, wouldn't it be better to start the timer when the player presses a button? You may think it's better to start the timer when the game says "Go," but personally I think starting when the player presses a button may give a better indication of their speed. I don't think this is a game-breaker, but this is just my 2 cents.
I think the current timer works better in the LongRun and Handicap mode though.
2. Make the game available on the Google Play Store. I see you have it available on App Store, so why not Google Play as well? I'm hazy on the specifics or how difficult this would be but it might give your game a wider audience.
I don't think much else needs changing other than maybe adding more game modes to make the game more interesting or introducing other animals, etc.