Game for Henry to master (5)
I believe the answer is:
halma
'game' is the definition.
(board game)
'henry to master' is the wordplay.
'henry' becomes 'hal' (short for Henry).
'to master' becomes 'ma' (Master of Arts).
'hal'+'ma'='HALMA'
'for' acts as a link.
(Other definitions for halma that I've seen before include "Board-game" , "Leaping board game" , "Hopping board game" , "Old board game" , "played on board" .)