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' says to put letters next to each other.
'sweetheart' becomes 'e' (the middle letter or heart of 'sweet').
'ma'+'e'='MAE'
'name' is the link.
This may not be right. It may be part of 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" .)