Hat produced around the Fourth of July (5)
I believe the answer is:
derby
'hat' is the definition.
(derby is a kind of hat)
'produced around the fourth of july' is the wordplay.
'produced' becomes 'bred' (breeding is a kind of producing).
'around' says the letters should be written in reverse.
'the fourth of july' becomes 'y' (fourth letter of the word 'July').
'bred' back-to-front is 'derb'.
'derb'+'y'='DERBY'
(Other definitions for derby that I've seen before include "20 horse 12" , "Race run at Epsom" , "Canadian bowler" , "English city" , "Flat race; hat" .)