Former wife, French and cuddling politician, offers excuse (6)
I believe the answer is:
exempt
'excuse' is the definition.
('exempt' can be a synonym of 'excuse')
'former wife french and cuddling politician' is the wordplay.
'former wife' becomes 'ex'.
'french and' becomes 'et' ('and' in the French language).
'cuddling' is an insertion indicator.
'politician' becomes 'mp'.
'et' enclosing 'mp' is 'empt'.
'ex'+'empt'='EXEMPT'
'offers' is the link.
(Other definitions for exempt that I've seen before include "Excused - let off" , "Free from, eg, tax" , "Not subject to an obligation" , "Duty-free" , "Not subject to or liable for" .)