Game, I sense, is draining for the finances (7)
I believe the answer is:
ruinous
'draining for the finances' is the definition.
(leading to financial ruin)
'game i sense' is the wordplay.
'game' becomes 'RU' (rugby union).
'sense' becomes 'nous' (nous is a kind of sense).
'ru'+'i'+'nous'='RUINOUS'
'is' is the link.
(Other definitions for ruinous that I've seen before include "Tumbledown - expensive" , "Calamitous, crippling" , "far too expensive" , "Catastrophic" , "Broken down" .)