This is just a guess, but I think it refers to when people are arguing back and forth. So if you're having a go, it's kind of like you're "having a turn" at criticizing that person?

Hi Akira, To 'go at someone' means to attack them. 'Have a go' is British English, and in British English, it's common to see 'have a...' For example: Have a bite = eat something; have a laugh = joke around / have fun.

Thank you for your comment jerry-san. In your example, "have a go" seems to mean just "do". So "have a go" includes all meaning of "do" depending on contexts? If so, it looks difficult for me to identify.( There is a word which works similarly in Japanese and it is difficult to explain how such word works to Japanese learners.)

This is not a BrE - only idiom and the meaning is not always about attacking someone or something. Ex: My friend is chopping down a tree but is too weak to do it. I can say: "Here, let me have a go at it.