The French annoyed by English game (8)
I believe the answer is:
lacrosse
'game' is the definition.
(lacrosse is a kind of game)
'the french annoyed by english' is the wordplay.
'the french' becomes 'la'.
'annoyed' becomes 'cross' (I've seen this before).
'by' means one lot of letters go next to another.
'english' becomes 'e' (abbreviation).
'la'+'cross'+'e'='LACROSSE'
(Other definitions for lacrosse that I've seen before include "Game played with a netted stick" , "Game played with nets on sticks" , "Game played with netted sticks" , "Ball game using netted stick" , "Score's uncertain after the Spanish" .)