What do I do in the summer after GCSEs?

Hey, I'm in year 11 and it's a big goal of mine to go to a uni like oxbridge. I have a conditional offer at Kings Maths School which I know has a high acceptance rate at oxbridge, and i've had this aim since year 8. Throughout secondary i've done sports (taekwondo) gone to competitions (mostly STEM and math based) and i'm part of my school's robotics team (which i'll have to leave after gcse 😞), but I don't know how to spend my summer in order to make my oxbridge application easier during sixth form. Teachers and older peers have suggested STEM summer schools, but i'm worried I haven't met the deadline and I can't afford most programs, also can't travel anywhere due to parental shared custody arrangements but i can't find any free online summer schools. Same issue for getting a summer job, with the travel, sort of. What do i do? Are there any things I haven't thought about that I can do over the summer to contribute to the application?

First of all, it's amazing that you got an offer for Kings Maths School. Well Done!!!
Second of all, Year 11 summer will be one of the last times where you'll get to fully enjoy yourself. Maybe learn a new skill/language or go out with friends.
What you have already done such as the STEM competitions and robotics club already shows interest now it keeping up with these commitments in year 12 because what Oxbridge truly cares about is if you truly have passion for the subject and you can demonstrate this in many ways such as reading books or watching lectures but this is stuff you'll be doing in year 12 where you'll learn knowledge that enables you to dig deeper. With summer schools, I'd say don't pay for any of them because unis tend not to like them and instead apply for ones like the Stem Smart one or the Sutton Trust ones that open up in year 12.
Tbh I didn't really even do any super curriculars for Oxbridge until year 12 summer but it's crazy that you're thinking about it now. You don't need crazy amounts of evidence. I had maybe one linked paragraph about aerogels research, a physics research project I did about invisibility cloaks, and a summer school at Imperial and I still somehow got in😭
What course are you thinking of applying to?
Honestly don't stress about this and focus on getting your GCSE grades and enjoy the summer. You could perhaps watch some Youtube vids or read books about stuff you like.

Thank you so much for the advice! I really appreciate it. To answer your question, I want to specialise in pure maths and/or potentially astrophysics at uni but I haven't set my mind on any particular course yet. I have heard of something called MMath at a bunch of unis which I find interesting and if possible I wanna take more of a researcher route. You mention having done physics research, how do you go about starting that if you don't mind me asking?

Hey, I'm in year 11 and it's a big goal of mine to go to a uni like oxbridge. I have a conditional offer at Kings Maths School which I know has a high acceptance rate at oxbridge, and i've had this aim since year 8. Throughout secondary i've done sports (taekwondo) gone to competitions (mostly STEM and math based) and i'm part of my school's robotics team (which i'll have to leave after gcse 😞), but I don't know how to spend my summer in order to make my oxbridge application easier during sixth form. Teachers and older peers have suggested STEM summer schools, but i'm worried I haven't met the deadline and I can't afford most programs, also can't travel anywhere due to parental shared custody arrangements but i can't find any free online summer schools. Same issue for getting a summer job, with the travel, sort of. What do i do? Are there any things I haven't thought about that I can do over the summer to contribute to the application?