OP -
It genuinely doesn’t matter if you’re a man or a woman. Your ex has moved on, remarried, and has another child. They have made their choices.
You should be moving on as well, and that means no joint therapy, etc. If you already pay for all the children’s expenses and all their clubs, and you’re the RP, and your ex refuses any help unless it’s on his or her terms… just enough. Stop. Don’t engage. Not with them. Not with the therapist. None of it. If your ex wants a different financial settlement, they can take you back to court. I very much doubt this will happen and even if it did, I doubt they’d be successful.
And to get very specific, no, when someone tells you that they want specific things (cinema tickets, etc) but then will only accept cash… I would absolutely refuse. You can use a voucher to go to any movie at any time at usually any one of a chain of theaters.
So please just extricate yourself from this whole situation with them, and stop questioning yourself. Your kids aren’t unhappy, and you shouldn’t be unhappy. Watch the situation just to make sure the ex is not badmouthing you to the children “oh, we can’t go anywhere because your mum/dad doesn’t love you enough to pay,” and otherwise, just ignore it. Good luck.