Demise admitting right's inadequacy (6)
I believe the answer is:
dearth
'inadequacy' is the definition.
(dearth is a kind of inadequacy)
'demise admitting right's' is the wordplay.
'demise' becomes 'death' ('death' can be a synonym of 'demise').
'admitting' is an insertion indicator.
'right' becomes 'r' (common abbreviation).
'death' placed around 'r' is 'DEARTH'.
(Other definitions for dearth that I've seen before include "Scarcity - lack" , "In short supply" , "Severe shortage" , "From shortage of food" , "Acute insufficiency" .)