Cool beat from the backing group (3,4)
I believe the answer is:
fan club
'the backing group' is the definition.
(group which backs a person or thing)
'cool beat' is the wordplay.
'cool' becomes 'fan' (fan can mean to cool down).
'beat' becomes 'club' (to hit with a club).
'fan'+'club'='FAN CLUB'
'from' acts as a link.
(Other definitions for fan club that I've seen before include "Organised group of admirers, as of pop stars" , "Fervent admirers" , "Association of devotees" , "Group devoted to a celebrity" , "Group of admirers as for pop star" .)