At the AGM John played the Chinese game (7)
I believe the answer is:
mahjong
'chinese game' is the definition.
(I've seen this before)
'at the agm john played' is the wordplay.
'at' says to put letters next to each other.
'played' indicates an anagram (letters playing around).
'agm' put after 'john' is 'johnagm'.
'johnagm' is an anagram of 'MAHJONG'.
'the' acts as a link.
(Other definitions for mahjong that I've seen before include "Game played with pieces called tiles" , "Chinese game using tiles" , "Chinese game with winds?" , "Game of Chinese origin played with tiles" , "Chinese tile game" .)