Heads of ambitious private schools excel in recess (4)
I believe the answer is:
apse
'recess' is the definition.
(apse is a kind of recess)
'heads of ambitious private schools excel' is the wordplay.
'heads of' indicates taking the first letters.
The initial letters of 'ambitious private schools excel' is 'APSE'.
'in' is the link.
(Other definitions for apse that I've seen before include "Semicircular end of a church, behind the altar" , "Where the altar is usually found in a church" , "where praetor sat" , "Peas (anag) -- part of a church" , "Architectural feature in a church" .)