Q Are governors responsible when the school is being used by the PTA or after school club?
A I will assume you mean responsible for any damage to persons or property outside school hours, and that you have in mind damage arising from the activity in question. This second point is important, since if any damage or injury occurs outside hours which is in no way connected with the letting, the school might well be held responsible if there had been negligence , for example, in failing to fill a hole or get a repair done.
On the assumption that damage or injury occurs as a result of the letting, and that there is no question of negligence on the school's part, the answer to your question depends on who is running the activity. If, for instance the after-school club is organised by the school, to take the simplest case, the school would be responsible and any claims would be on the school insurance.
If the LEA is involved it would be responsible either wholly or in association with the school. If it is a club organised independently by a third party the organisers should ensure that their own insurance covers any possible damage or injury, since the school will not be liable (except, as I have said, where negligence is proved).
Parents' associations normally have insurance which covers all their activities, either through their membership of the NCPTA or independently. However where the occasion is one of the parents' free lettings it may be that the school's insurance will cover it.
It is always difficult to answer questions about insurance with any precision because local authorities and schools' policies do vary quite a lot, and so do the letting terms for school premises. It is important that governors should keep their insurance situation under review, including conditions applying to lettings. They should always pay attention to what is agreed with groups renting school premises after hours and make sure that responsibility is clear to both parties.