Moderately shrewd about the broadcasting medium (6)
I believe the answer is:
fairly
'moderately' is the definition.
('fairly' can be a synonym of 'moderately')
'about the broadcasting medium' is the wordplay.
'about' means one lot of letters goes inside another.
'the broadcasting' becomes 'fly' (I can't explain this - if you can you should give a lot more credence to this answer).
'medium' becomes 'air' (air is a kind of medium).
'fly' placed around 'air' is 'FAIRLY'.
'shrewd' is the link.
This may not be right. It may belong to another bit of the clue.
Can you help me to learn more?
(Other definitions for fairly that I've seen before include "In an evenhanded manner - somewhat" , "pretty" , "Quite" , "Moderately" , "Evenhandedly" .)