Sounds like they're suffering. In their chests? (7)
I believe the answer is:
coffers
'chests?' is the definition.
(coffer is a kind of chest)
I cannot understand how the remainder of the clue works.
Can you help me to learn more?
(Other definitions for coffers that I've seen before include "Money-chests" , "Treasury boxes" , "Treasure chests" , "Strongboxes" , "Supply of funds" .)