Run with a cry into the street (5)
I believe the answer is:
scoot
'run' is the definition.
(thesaurus)
'a cry into the street' is the wordplay.
'a cry' becomes 'coo' (coo is a kind of cry**).
'into' means one lot of letters goes inside another.
'the street' becomes 'st'.
'coo' inserted inside 'st' is 'SCOOT'.
'with' 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)" .)