I’m a single mum and reduced my hours to one day a week last year. I asked for a set day, but was told the day would be a random one and that I would have to work one weekend in four. In the event, I was only asked to work weekends twice in the whole year. This year I have put in another flexible working request to get my one day a week fixed so I can manage childcare. This was denied. My starting hours have now been changed to earlier and I have been asked to work on some Saturdays. I have no childcare on weekends at all. I feel that they are deliberately trying to force me out because they know that.
Unfortunately, you can only really go with what your contract states legally. You could say that over the last year you haven’t been working regular Saturdays and this could be seen as an implied term. It would be worth raising it formally with HR.
You could ask instead of Saturdays that you cover someone else’s day in the week, so a swap for working the Saturday. Other than that it would be another flexible working request, 12 months after the first one, or looking for another role in the businesses that doesn’t include working on Saturdays.