Chief sounds like horse hair (4)
I believe the answer is:
main
'chief' is the definition.
(I know that chief can be written as main)
'sounds like horse hair' is the wordplay.
'sounds like' shows a homophone (sound like).
'horse hair' becomes 'mane' (I've seen this in another clue).
'mane' sounds like 'MAIN'.
(Other definitions for main that I've seen before include "Most important - high seas?" , "location for 9 [ARMADA]" , "associated with the Spanish?" , "river in Germany" , "Foremost" .)