Game one's put down after a couple of drinks (3,5)
I believe the answer is:
gin rummy
'game' is the definition.
(card game)
'one's put down after a couple of drinks' is the wordplay.
'a couple of drinks' can mean two replacements for 'drink'.
'one's' becomes 'my' (both can mean belonging to the person writing).
'put down after' means one lot of letters go next to another (some letters go after others).
'drink' becomes 'gin' (gin is a kind of drink).
'drink' becomes 'rum' (rum is a kind of drink).
'gin'+'rum'='ginrum'
'my' after 'ginrum' is 'GIN RUMMY'.
(Other definitions for gin rummy that I've seen before include "cards" , "Card game" .)