Tenacious for every first broadcast outside (10)
I believe the answer is:
persistent
'tenacious' is the definition.
(I know that tenacious can be written as persistent)
'for every first broadcast outside' is the wordplay.
'for every' becomes 'per' ('per' means 'for each' or 'for every').
'first' becomes 'ist' (ist resembles sst).
'broadcast' becomes 'sent' (I've seen this in other clues).
'outside' indicates putting letters inside.
'ist' going within 'sent' is 'sistent'.
'per'+'sistent'='PERSISTENT'
(Other definitions for persistent that I've seen before include "Continual" , "hanging on" , "fixed" , "Prolonged; nagging" , "single-minded" .)