Girl having wager with husband (4)
I believe the answer is:
beth
'girl' is the definition.
'wager with husband' is the wordplay.
'wager' becomes 'bet' ('bet' can be a synonym of 'wager').
'with' is a charade indicator (letters next to each other).
'husband' becomes 'h'.
'bet'+'h'='BETH'
'having' is the link.
(Other definitions for beth that I've seen before include "Hebrew letter; a Little Woman" , "Girl" , "Little Women sister" , "letter to the Hebrews" , "Hebrew B" .)