Advice please - I need to approach work about asking for flexibility in school holidays

I’d check if you have some sort of flexible working policy at work (most uk companies in the uk do) and go from there. I have been a manager for over a decade and one thing that never worked are verbal agreements, they tend to get forgotten when new managers start, not handed over , not kept etc.

Does your workplace allow time off in lieu (TOIL)?

Are you able to wfh and do your hours when your child has gone to bed?

All you can do is ask and anything you agree between you make sure you get formally agreed in writing.
They can refuse of course. We just use holiday and holiday clubs to get through the holidays!

Yeah. We have to do 3 weeks of clubs! Then take holiday for the rest