Graduate takes ages, heading off for period around arrival time (9)
I believe the answer is:
maternity
'period around arrival time' is the definition.
(period around the arrival of a baby)
'graduate takes ages heading off' is the wordplay.
I cannot quite understand how this works, but
'graduate' could be 'ma' (abbreviation for Master of Arts) and 'ma' is found within the answer.
This may be the basis of clue (or it may be nonsense).
'for' is the link.
(Other definitions for maternity that I've seen before include "Dad never gets it" , "Motherhood" , "Ward" .)