Booby possibly to attempt crossing river (4)
I believe the answer is:
bird
'booby possibly' is the definition.
(booby is a kind of bird)
'attempt crossing river' is the wordplay.
'attempt' becomes 'bid' (bid is a kind of attempt).
'crossing' indicates putting letters inside (to cross can mean to go over or around).
'river' becomes 'r' (abbreviation).
'bid' enclosing 'r' is 'BIRD'.
'to' is the link.
(Other definitions for bird that I've seen before include "Booby, for example" , "Nest builder?" , "that flies" , "Prison time (slang)" , "Avian" .)