Former partner, constant, embarrassed, having run out (7)
I believe the answer is:
expired
'run out' is the definition.
(I've seen this before)
'former partner constant embarrassed' is the wordplay.
'former partner' becomes 'ex' (an ex-partner).
'constant' becomes 'pi' (mathematical constant).
'embarrassed' becomes 'red' (someone may blush or go red when embarrassed).
'ex'+'pi'+'red'='EXPIRED'
'having' acts as a link.
(Other definitions for expired that I've seen before include "Lapsed" , "Run out" , "Ran out or died" , "Became invalid" , "Came to an end" .)