One clearing throat loudly? It's the chest (6)
I believe the answer is:
coffer
'the chest' is the definition.
(coffer is a kind of chest)
'one clearing throat loudly?' is the wordplay.
'one clearing throat' becomes 'cougher'.
'loudly?' indicates a 'sounds like' (homophone) clue (I've seen 'loudly' mean this).
'cougher' sounds like 'COFFER'.
'it's' is the link.
(Other definitions for coffer that I've seen before include "jewellery store?" , "Treasury - chest for valuables" , "Strongbox" , "A large strong box for valuables" , "safe" .)