Go one better than catching river fish (4)
I believe the answer is:
carp
'fish' is the definition.
(carp is a kind of fish)
'go one better than catching river' is the wordplay.
'go one better than' becomes 'cap' (to cap someone is to beat or better them).
'catching' means one lot of letters goes inside another.
'river' becomes 'r' (abbreviation for river).
'cap' enclosing 'r' is 'CARP'.
(Other definitions for carp that I've seen before include "ish" , "Cavil - fish" , "Swimmer" , "Continuously find fault" , "Pond fish" .)