Former partner is given short time to live (5)
I believe the answer is:
Here is my best explanation:
'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).
'to' is the link.
(Other definitions for exist that I've seen before include "Subsist", "Be alive", "quick!", "Have life", "Scrape by", "TO BE", "Have objective reality", "Be, live", "Have being or continue living" and "Are".)