My current client had a non-compete in their contract specifying I would not be allowed to work for one of their customers, or something like that. This client is one of the big four accountancy firms; their customers can really include everyone, and I have no way to tell who is or isn't their customer.

And my work doesn't even include any contact with their customers! What they really wanted was that I wouldn't do for, or even share with, the other big accountancy firms what I was going to do for them, because it's something unusual that's their idea and they think it gives them an advantage. That's completely fine of course. The original reading was ridiculous.

I don't think they changed it in the contract, but they did explain it by email.

That sounds like it might be an audit restriction. Those places have very strict rules separating who they do accounting or consulting for vs who they audit, for obvious reasons. So it might be that they’re saying you can’t do work for them in one capacity and then go work for somebody they audit.