That girl's father is half cut (3)
I believe the answer is:
her
'that girl's' is the definition.
('her' refers to a girl or woman)
'father is half cut' is the wordplay.
'is half cut' means to take half the letters.
'father' halved is 'HER'.
(Other definitions for her that I've seen before include "Belonging to the woman" , "In the Midlands she" , "That lady's" , "Pronoun used of ship" , "She, objectively" .)