Expose story, essentially controversial, Greek character rejected (6)
I believe the answer is:
unveil
'expose' is the definition.
(unveiling is a kind of exposing)
'story essentially controversial greek character rejected' is the wordplay.
'story' becomes 'lie' (story can mean a fabrication).
'essentially' says to take the centre.
'greek character' becomes 'nu' (letter from the Greek alphabet).
'rejected' shows that the letters should be reversed in order.
The central letter of 'controversial' is 'v'.
'lie'+'v'+'nu'='lievnu'
'lievnu' in reverse letter order is 'UNVEIL'.
(Other definitions for unveil that I've seen before include "bare" , "Introduce publicly for the first time" , "Show or announce publicly for the first time" , "Expose or reveal, remove cover" , "Remove headdress" .)