Flyer’s broadcast gripping sixth wife (7)
I believe the answer is:
sparrow
'flyer's' is the definition.
(type of bird)
'broadcast gripping sixth wife' is the wordplay.
'broadcast' becomes 'sow' (sowing is a kind of broadcasting).
'gripping' means one lot of letters goes inside another.
'sixth wife' becomes 'Parr' (Catherine Parr, sixth wife of Henry VIII).
'sow' going around 'parr' is 'SPARROW'.
(Other definitions for sparrow that I've seen before include "Small songbird" , "Passerine bird" , "a British resident" , "Who killed Cock Robin" , "Cock Robin's assassin" .)