Henry's half cut! (3)
I believe the answer is:
hal
'henry's' is the definition.
(short for Henry)
'half cut' is the wordplay.
'cut' means to remove the last letter.
'half' with its last letter removed is 'HAL'.
(Other definitions for hal that I've seen before include "Henry (diminutive)" , "future Henry V" , "Falstaff's Prince; 2001 computer" , "Falstaff's prince (H.IV)" , "Prince -, Falstaff's companion (Henry IV, Pt I)" .)