The master has sweetheart name her! (3)
I believe the answer is:
mae
'her' is the definition.
'the master has sweetheart' is the wordplay.
'the master' becomes 'ma' (Master of Arts).
'has' is a charade indicator (letters next to each other).
'sweetheart' becomes 'e' (the heart/centre of 'sweet').
'ma'+'e'='MAE'
'name' acts as a link.
I am not very happy about this link. It may belong to another bit of the clue.
Can you help me to learn more?
(Other definitions for mae that I've seen before include "The first Hollywood blonde femme fatale, . . . West" , "Girl" , "Woman" , "- - West, movie vamp" , "- - West, life-jacket" .)