She's desperately after clubs in game (5)
I believe the answer is:
chess
'game' is the definition.
(chess game is a kind of game)
'she's desperately after clubs' is the wordplay.
'desperately' indicates anagramming the letters.
'after' says to put letters next to each other.
'clubs' becomes 'c' (abbreviation used in card games such as bridge).
'shes' is an anagram of 'hess'.
'hess' after 'c' is 'CHESS'.
'in' is the link.
(Other definitions for chess that I've seen before include "Board game of 13, 16 and 21" , "form of battle" , "Ancient war game whose name may have derived for the Persian for 'king'" , "Game involving mates" , "A game of skill" .)