I don't recall any of the Hawks having "F" designation, they are "T" - trainers in RAF use, not fighters (arguably DFGA with the gun and some little bombs) - although with AIM9 attached (and of course the Aden cannon) they could fill a minor A-A defensive/support role.
6000' is ample (in the training fit); on a hot day with no wind and a heavier payload, you might want a bit more.
Mister B