Former partner is given short time to live (5)
I believe the answer is:
exist
'live' is the definition.
('exist' can be a synonym of 'live')
'former partner is given short time' is the wordplay.
'former partner' becomes 'ex'.
'given' is a charade indicator (letters next to each other).
'short time' becomes 't' (abbreviation for time).
'ex'+'is'+'t'='EXIST'
'to' is the link.
(Other definitions for exist that I've seen before include "Have being or continue living" , "Have objective reality" , "Get a life!" , "quick!" , "Live" .)