Race for a hat (5)
I believe the answer is:
derby
I believe this is a double definition.
'race' is the first definition.
(I've seen this before)
'for a hat' is the second definition.
(derby is a kind of hat)
(Other definitions for derby that I've seen before include "20 horse 12" , "Local game" , "Annual race at Epsom" , "Sports match between local rival teams" , "Type of hat" .)