Country club left out American (4)
I believe the answer is:
cuba
'country' is the definition.
(Cuba is an example)
'club left out american' is the wordplay.
'left' becomes 'l' (common abbreviation).
'out' indicates named letters should be taken away.
'american' becomes 'a' (common abbreviation - e.g. in organisation names).
'club' with 'l' taken away is 'cub'.
'cub'+'a'='CUBA'
(Other definitions for cuba that I've seen before include "Country famous for its cigars" , "Large island discovered by Columbus, 1492" , "Castro's republic" , "Its capital is Havana" , "tropical country" .)