Jack round with a new woman (4)
I believe the answer is:
joan
'woman' is the definition.
'jack round with a new' is the wordplay.
'jack' becomes 'j' (playing card abbreviation).
'round' becomes 'o' (round can mean a round object**).
'with' means one lot of letters go next to another.
'a new' becomes 'an' (I've seen this in other clues).
'j'+'o'+'an'='JOAN'
(Other definitions for joan that I've seen before include "- - Collins, actress" , "Darby and --, -- of Arc" , ". . . . of Arc, French warrior saint" , "Orleans girl" , "- Sutherland, Australian soprano" .)