Grasped by one of those at death's door with chesty problem (6)
I believe the answer is:
scythe
'grasped by one of those at death's door' is the definition.
I can't tell whether this definition defines the answer.
'chesty problem' is the wordplay.
'problem' is an anagram indicator.
'chesty' with letters rearranged gives 'SCYTHE'.
'with' is the link.
Can you help me to learn more?
(Other definitions for scythe that I've seen before include "Old mowing implement for hand use" , "Cutting instrument" , "Old farming tool" , "Implement with curved blade" , "Long-handled cutter" .)