Translation:Please give me a chance.
Please give me one more chance = 请再给我一个机会
再 means again. 请再给我一个机会 = Please give me a chance again = Please give me one more chance.
Even though the meaning is very sound here, Chinese and English do not have the same (and obvious) ways of expressing sentences.
Since you are emphasizing "one more", this translates the classifier to 次 instead of 个; otherwise, you are simply specifying "one chance".