Prevail as headless chauvinist (5)
I believe the answer is:
exist
'prevail' is the definition.
(I know that prevail is a kind of exist)
'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 initial letter taken away is 'EXIST'.
'as' is the link.
(Other definitions for exist that I've seen before include "Live" , "Have being or continue living" , "Have objective reality" , "Scrape by" , "Subsist" .)