Exhausted, catching eastern river snake (7)
I believe the answer is:
serpent
'snake' is the definition.
(I know that snake can be written as serpent)
'exhausted catching eastern river' is the wordplay.
'exhausted' becomes 'spent' (similar in meaning).
'catching' indicates putting letters inside.
'eastern' becomes 'e'.
'river' becomes 'r' (abbreviation for river).
'e'+'r'='er'
'spent' enclosing 'er' is 'SERPENT'.
(Other definitions for serpent that I've seen before include "exhibit at zoo" , "Present for a snake" , "Instrument of the past" , "(In literature) a snake" , "Present (anag.)" .)