Short letter from school turned up (4)
I believe the answer is:
note
'short letter' is the definition.
('note' can be a synonym of 'short letter')
'school turned up' is the wordplay.
'school' becomes 'eton' (famous UK public school).
'turned up' is a reversal indicator.
'eton' backwards is 'NOTE'.
'from' is the link.
(Other definitions for note that I've seen before include "Short communication" , "Fame - memorandum" , "Eg, fiver; annotation" , "A piece of paper money" , "Traitor" .)