Not an obtuse question but what exactly is the gap you want bridged?
There isn't automatically one just because it's a state school therefore if you perceive there to be one you first need to identify if.
I think private school kids are, in general, putting in more hours of work. They may get a couple of weeks extra holiday in the summer, but this is more than counterbalanced by (in general) longer school days, more homework, more weekend work, more holiday work. That's why I would personally recommend doing some extra work outside of school, esp. during the holidays which is when kids tend to slip back
Some kids read and write a lot off their own bat, so are likely to do well in the language arts regardless of what type of school they go to and regardless of what their parents do. That was me, incidentally--comprehensive student, went on to Cambridge. I think I would have done better in maths if my parents had pushed me more and had me tutored, though.
(Disclaimer: I am talking about standard academically-focused private schools. There are different types of private schools, of course, including some that pride themselves on being very alternative and child-led and so on.)
You know what I find myself wondering, though? You hear a LOT of parents saying that they decided to send their children to private school because of the sport and music options.
(I don't know whether they are being a bit disingenous and really it was better exam results they were hoping for but felt a bit embarassed about saying out loud... but, OK, so private schools usually have better music and sports facilities.)
So:
What happens if you send your kid to a private primary school with all these glorious facilities (requiring glorious fees, needless to say, to cover all the costs), and then your kid turns out to be interested in neither? I went to a state primary school that had a good orchestra and unusually good sporting facilities (and lots of emphasis on sport).
I hated the sport with a passion, and despite being approached and asked to take up an orchestra instrument, made it clear that the only thing I wanted to play was the piano. By myself. With the neighborhood piano teacher lady who lived down the road.
My parents, of course, didn't mindno skin off their noses, since this was a state school anyway. But I'm just trying to imaginehow would they have felt if they were stuck paying $$$$ a year for all these wonderful facilities and their child simply wasn't interested? I can't imagine how frustrating that would be.
I think it makes sense, in most cases, to pay for extra curriculars outside of school , since this means you can opt in and out and you don't have to pay for anything that your child doesn't want to do.