Slip right into East River (3)
I believe the answer is:
err
'slip' is the definition.
(to err is to make an error)
'right into east river' is the wordplay.
'right' becomes 'r' (common abbreviation).
'into' means one lot of letters goes inside another.
'east' becomes 'e' (synonyms).
'river' becomes 'r' (abbreviation for river).
'e'+'r'='er'
'r' placed inside 'er' is 'ERR'.
(Other definitions for err that I've seen before include "'to... is human,to forgive divine'" , "Be mistaken" , "Bungle" , "Stray from the straight and narrow" , "Be incorrect" .)