Service's push following broadcast (3,5)
I believe the answer is:
air force
'service's' is the definition.
(air force is a kind of service)
'push following broadcast' is the wordplay.
'push' becomes 'force' ('force' can be a synonym of 'push'**).
'following' says to put letters next to each other.
'broadcast' becomes 'air' ('air' can be a synonym of 'broadcast').
'force' after 'air' is 'AIR FORCE'.
(Other definitions for air force that I've seen before include "Junior service" , "Country's military planes" , "one of the services" , "Armed service" , "A country's flying service" .)