Yes. As Laura Jane Walker says, this colloquial expression is common in British English. "She's always having a go at me" is a very natural, informal way of saying that someone frequently criticises or reprimands you. You will often hear this in the UK. NB Grammar note: This is also a very natural example of a present continuous used with 'always' to describe repeated behaviour which the speaker sees as excessive or unreasonable.

As Kassie says, it can mean either "to criticise", or just "to try to do something". If you add "at someone" then it's criticizing, if "at something" then it's trying to do it. Confusing!

In the US “have a go” is like “to take a chance” or “to try to do something” so like if someone is trying to do something without any success, someone else shows up and says “let me have a go” In British English, it’s what you say!

It’s not super common here in the US but I have definitely heard it. I think how much it is used depends on what part of the English speaking world your in. Regardless, It would be colloquial. Here it’s more common that expression in other ways that don’t mean “to criticize” like “to make an attempt”. Example “Let me have a go at that problem.”

Yes, me too. I'm sure everyone does (I'm British as well). It's maybe worth noting that, if you want to use it, in the negative you can drop the "at x" bit. "Look, I'm not having a go, I just want to help you find your phone," for example.