Live like a headless chauvinist (5)
I believe the answer is:
exist
'live' is the definition.
(synonyms)
'headless chauvinist' is the wordplay.
'headless' means to remove the first letter.
'chauvinist' becomes 'sexist' (I've seen this in another clue).
'sexist' with its first letter taken away is 'EXIST'.
'like a' is the link.
(Other definitions for exist that I've seen before include "Subsist" , "Get a life!" , "Have objective reality" , "quick!" , "Survive" .)