Run with a cry into the street (5)
I believe the answer is:
scoot
'run' is the definition.
(I've seen this in another clue)
'cry into the street' is the wordplay.
'cry' becomes 'coo' (coo is a kind of cry).
'into' is an insertion indicator.
'the street' becomes 'st' (abbreviation).
'coo' placed inside 'st' is 'SCOOT'.
'with a' is the link.
(Other definitions for scoot that I've seen before include "Move rapidly" , "Move with a rapid darting motion" , "hurry" , "Dart off" , "Depart quickly (does not contain H)" .)