Somehow not mind following graduate game (9)
I believe the answer is:
badminton
'game' is the definition.
(badminton is a kind of game)
'somehow not mind following graduate' is the wordplay.
'somehow' is an anagram indicator.
'following' is a charade indicator (letters next to each other).
'graduate' becomes 'BA' (someone with a Bachelor of Arts degree).
'not'+'mind'='notmind'
'notmind' with letters rearranged gives 'dminton'.
'dminton' put after 'ba' is 'BADMINTON'.
(Other definitions for badminton that I've seen before include "One sport" , "22 Across sport" , "Game with shuttlecocks" , "in which rackets are involved" , "Seat of the Duke of Beaufort - racket game" .)