Clearly you'll have to tread carefully, which means treating advice sourced from the internet with some care!
My way of thinking of things when someone requests work for free (or if they might): You pay me for my hours, and you leave to my goodwill and discretion what I throw in for free – you don't request nor demand my goodwill.
If someone does the latter, I bill them in full for the hours they have requested. And then I decide if I still throw in anything for free.
Ultimately, good clients will allow me work like this (implicitly), bad clients will want to 'win' at every turn.
Maybe I'd try suggesting some version of the philosophy I express above; I did so with a client recently who was trying to get stuff out of me for free. He was charged in full but I threw in easily 2-3 times the value in extras. There was no fall-out and the contract is still active.
Ultimately this requires pushback in a smart way, and pushback now imo, otherwise the client establishes a hold over you which in future you will find much harder than this to escape.
I don't know your approach to providing value, but you must be confident that you provide it, and so I would work your pushback / response around that.