America backed dodgy gaffer in vote (8)
I believe the answer is:
'vote' is the definition.
'america backed dodgy gaffer' is the wordplay.
'america' becomes 'U.S.' (abbreviation for the United States).
'backed' says the letters should be written backwards.
'dodgy' is an anagram indicator.
'us' back-to-front is 'su'.
'gaffer' anagrammed gives 'ffrage'.
'in' acts as a link.
(Other definitions for suffrage that I've seen before include "The right to vote in political elections", "The right to vote, as won by women".)