Person with ticklish throat picked up something chesty? (6)
I believe the answer is:
coffer
'something chesty?' is the definition.
(a coffer is a chest or trunk)
'person with ticklish throat picked up' is the wordplay.
'person with ticklish throat' becomes 'cougher' (someone coughing).
'picked up' shows a homophone (sound like).
'cougher' is a homophone of 'COFFER'.
(Other definitions for coffer that I've seen before include "Chest for storing valuables" , "Treasury - chest for valuables" , "Strongbox" , "safe" , "A large strong box for valuables" .)