Girl has wager with Henry (4)
I believe the answer is:
beth
'girl' is the definition.
(girl's name)
'wager with henry' is the wordplay.
'wager' becomes 'bet' (synonyms).
'with' means one lot of letters go next to another.
'henry' becomes 'H' (symbol for the unit of electrical inductance).
'bet'+'h'='BETH'
'has' acts as a link.
(Other definitions for beth that I've seen before include "Concluding stanza" , "Meg, Jo, - - and Amy" , "Girl" , "Hebrew B" , "Little Women sister" .)