Cad with that woman sooner (6)
I believe the answer is:
rather
'sooner' is the definition.
('I'd sooner do this' means 'I'd rather do this')
'cad with that woman' is the wordplay.
'cad' becomes 'rat' (I've seen this before).
'with' means one lot of letters go next to another.
'that woman' becomes 'her' (her refers to a woman).
'rat'+'her'='RATHER'
(Other definitions for rather that I've seen before include "To a certain extent" , "Fairly" , "More precisely; instead of" , "with more justification" , "Slightly, more so than not (6" .)