Run out of Gold Street going after former husband (7)
I believe the answer is:
exhaust
'run out of' is the definition.
('exhaust' can be a synonym of 'run out of')
'gold street going after former husband' is the wordplay.
'gold' becomes 'au' ('Au' can be a synonym of 'gold').
'street' becomes 'st'.
'going after' is a charade indicator (letters next to each other).
'former' becomes 'ex'.
'husband' becomes 'h'.
'au'+'st'='aust'
'ex'+'h'='exh'
'aust' after 'exh' is 'EXHAUST'.
(Other definitions for exhaust that I've seen before include "Deplete" , "Waste gas from engine" , "Wear, tire out" , "Tire utterly or fumes-dispersing device in car" , "Sap - piped emission" .)