Game found by river — deep but not wide (4)
I believe the answer is:
polo
'game' is the definition.
(polo is a kind of game)
'by river deep but not wide' is the wordplay.
'by' means one lot of letters go next to another.
'river' becomes 'po' (Po is an example).
'deep' becomes 'low' ('low' can be similar in meaning to 'deep').
'but not' suggests deleting specific letters.
'wide' becomes 'w' (abbreviation e.g. in cricket).
'low' with 'w' taken away is 'lo'.
'po'+'lo' is 'POLO'.
'found' is the link.
(Other definitions for polo that I've seen before include "Game - Venetian traveller" , "Game played on horses or in water" , "Game played on horseback wielding mallets" , "Oriental visitor < ? Sucker having a hole" , "Game; Italian explorer" .)