Pigeon in river (5)
I believe the answer is:
homer
'pigeon' is the definition.
(homer is a kind of pigeon)
'in river' is the wordplay.
'in' becomes 'home' ('he's in' can mean 'he's home').
'river' becomes 'r'.
'home'+'r'='HOMER'
(Other definitions for homer that I've seen before include "Bird" , "trained to get back to house" , "successful run in America?" , "Greek poet, c. 800 BC" , "Bart Simpson's father" .)