take them to museums and concerts, insist on supper at the table every night, expose them to culture and history, who show their kids that the library is exciting
Although whether that's a cause or an effect is an interesting research project. If you gave single parents who left school at sixteen with few qualifications, living in deprived areas, tickets and transport to the RSC once a month, would there be a net benefit to their children? There might be. But it might be that the real benefit to children of theatre going (as an example) occurs on the way there and back, with parents who can use Richard II as a teachable moment, and those parents are finding those teachable moments (and being able to exploit them) in many more places than visits to the theatre.
You could probably equally observe that children who do well at GCSE are disproportionately likely to wear, and have parents that wear, Boden, but distributing vouchers for sensible clothes in natural fibres to underachieving schools seems a rather indirect way to boost 5 A*-C results.
And culture isn't free. Aside from the cost of tickets, which may be free, there's also the transport cost of getting to the event, the opportunity cost of not working, the lunch you need, etc. There's a slight whiff of "let them eat cake" to the proposition that just because the main collection at the National Gallery is free, single parents in Doncaster are remiss for not showing their children more Van Gogh, and I'm not sure what in isolation that visit would achieve. Cultural capital is about a lot more than standing in the room, and the reasons why children who have backgrounds that value culture tend to do better at school is a complex mesh of causes and effects.