A vote beginning to offend God (6)
I believe the answer is:
apollo
'god' is the definition.
(Apollo is a Greek god)
'a vote beginning to offend' is the wordplay.
'vote' becomes 'poll' (polling is a kind of voting).
'beginning to' suggests taking the first letters.
The first letter of 'offend' is 'o'.
'a'+'poll'+'o'='APOLLO'
(Other definitions for apollo that I've seen before include "Greek god - US space programme dedicated to moon landings - butterfly" , "source of name for far-flung missions" , "Greek god of the sun (6)" , "handsome chap" , "US space project, 1960-72" .)