Schools like this no longer exist in Britain, but we used to have what were termed "direct-grant schools". Local education authorities used to pay grants to these schools but these did not cover all the costs and so most pupils at "direct-grant schools" had to pay fees, though not as much as at a truly private school. I think Brits would understand "state-susidised" to mean state schools. Public-private partnerships usually involve businesses.
Another possibility would be "grant-aided". I think Brits would understand this to mean that some funds are provided by the government but that not all the costs are covered.
Yes, I do understand you explanation Soop Puss, though here in Spain you can take your children to such "colegio concertado" and ask for a grant as well, so I think the terms "direct-grant school" or "grant-aided" would be confused.
What about "state financed but private management school"?
Perhaps I wasn't clear: the grant went to the school, not to the pupil. Grants were not calculated on the basis of individual pupils but as a total amount depending on the views of the local education authority: perhaps they disagreed with the concept of "direct-grant schools" and so would give less grant; perhaps they wanted to encourage the school to take in more children from families that couldn't afford to pay the fees and so would give more grant; etc.
I still think "state-financed but privately-managed" suggests all the costs are covered by the government. And in fact, "privately managed" even suggests the school is owned by the education authority but they have subcontracted the management to a private company or organisation.
From what you say, I understand that your translation is intended for Spanish readers, rather than native English speakers. That being the case, perhaps you should ask a few Spanish speakers what they understand by tpmonline's suggestion "state-subsidised". If they think the term describes a "colegio concertado", that will probably be the best term for you to use.