I'm not sure how to say it in English. The baseball game has 9 innings, doesn't it? 回が浅い means "a few innings of the first parts of the game" like the first, second, third inning and the speaker said that it's easier to hit the ball (to take his turn) in those innings because of having less pressure comparing with that of the 8th or 9th inning. I hope you understand more than before.