Musical professorship is a non-starter (4)
I believe the answer is:
hair
'musical' is the definition.
(1960s musical)
'professorship is a non-starter' is the wordplay.
'professorship' becomes 'chair' (synonyms).
'is a non-starter' means to remove the first letter (the letter which starts the word is removed).
'chair' with its first letter taken off is 'HAIR'.
(Other definitions for hair that I've seen before include "Fur - mop" , "It helps prevent heat loss through the head" , "Famous hippy musical of hirsute character" , "Tresses" , "Outgrowth from the skin" .)