Graduate in French department has work for early morning (6)
I believe the answer is:
aubade
'work for early morning' is the definition.
(poem or piece of music for early morning)
'graduate in french department' is the wordplay.
'graduate' becomes 'BA' (someone with a Bachelor of Arts degree).
'in' means one lot of letters goes inside another.
'french department' becomes 'aude' (Aude is one of the departments of France).
'ba' inserted inside 'aude' is 'AUBADE'.
'has' is the link.
(Other definitions for aubade that I've seen before include "Piece of music appropriate to the dawn" , "Early song" , "Early announcement" , "Dawn music or poem" , "Sunrise song" .)