District gets prize, but not the first one (4)
I believe the answer is:
ward
'district' is the definition.
(ward is a kind of district)
'prize but not the first one' is the wordplay.
'prize' becomes 'award' ('award' can be a synonym of 'prize').
'but not the first one' means to remove the first letter.
'award' with its first letter taken off is 'WARD'.
'gets' is the link.
(Other definitions for ward that I've seen before include "Separate room" , "Councillor's area" , "Electoral district; Mrs Humphry ---" , "treatment room" , "Local administrative district" .)