Nag Republican to wear headgear (4)
I believe the answer is:
carp
'nag' is the definition.
(I've seen this in another clue)
'republican to wear headgear' is the wordplay.
'republican' becomes 'r' (abbreviation).
'to wear' is an insertion indicator (to wear something is to be covered up by it).
'headgear' becomes 'cap' (cap is a kind of headgear).
'r' put into 'cap' is 'CARP'.
(Other definitions for carp that I've seen before include "Complain in manner of goldfish" , "Fish - grouse" , "Fish - complaint" , "Complain continually" , "Moan; fish" .)