Reportedly throw away covering of sleeveless garment (9)
I believe the answer is:
waistcoat
'sleeveless garment' is the definition.
(I've seen this before)
'reportedly throw away covering' is the wordplay.
'reportedly' shows a homophone (sound like).
'throw away' becomes 'waste' (wasting is a kind of throwing away).
'covering' becomes 'coat' (coat is a kind of covering).
'waste' sounds like 'waist'.
'waist'+'coat'='WAISTCOAT'
'of' acts as a link.
(Other definitions for waistcoat that I've seen before include "Part of a three-piece suit" , "American's vest" , "Garment partly visible under a jacket" , "Something worn" , "A "vest" in the US" .)