Because of that, their farewell's on radio (7)
I believe the answer is:
thereby
'because of that' is the definition.
(I've seen this before)
'their farewell's on radio' is the wordplay.
'their' becomes 'there' (I am not sure about this - if you are sure you should believe this answer much more).
'farewell' becomes 'bye' (bye is a kind of farewell).
'on radio' indicates a 'sounds like' (homophone) clue.
'bye' is a homophone of 'by'.
'there'+'by'='THEREBY'
Can you help me to learn more?
(Other definitions for thereby that I've seen before include "In consequence" , "about that amount" , "By that means" , "For that reason" , "As a result of that" .)