This is a harder problem than just "Free Education".
I grew up in the UK (in "deprived" areas, to a single parent with no income other than jobseekers allowance and some childcare from the state) and I can formerly attest that my "free education" was atrocious.
I am not sure if there was a negative feedback loop involved: School funding was determined by efficacy. How do you measure efficacy of a school? Well, test results... which people from this kind of background will not have good ones.
Another issue that perhaps is not thought much about is that 'Homework' for impoverished people is very difficult, in my case I didn't have a place to actually sit and work (no dining table, we ate on our laps). But Homework is a necessary part of education.
Add to the fact that socio-economic factors do filter into the school; things like gangs and in-groups.
Worried about gangs and gang colours: uniform
Uniforms cost money. Money people don't have.
By the time you get to the age where you're allowed to work (or go to university)- you take what you can get to get out of poverty; which is not going to be further education.
In my experience school is a temporary prison for children so that their parents can go to work, I definitely didn't learn the things I know from school. I learned them from my mum and school regurgitated them at me ad infinitum in a boring classroom in a small building surrounded by prison bars.
I do not miss free school. I strongly resent that it was ever forced on me.