Lacking function, bodyguard is possible cat-killer (8)
I believe the answer is:
nosiness
'possible cat-killer' is the definition.
('curiosity killed the cat')
'lacking function bodyguard' is the wordplay.
'lacking' becomes 'no' (eg lacking money means with no money).
'function' becomes 'sine' (mathematical function).
'bodyguard' becomes 'ss' (the SS was originally Hitler's personal bodyguard).
'no'+'sine'+'ss'='NOSINESS'
'is' acts as a link.
(Other definitions for nosiness that I've seen before include "Tendency to pry" , "Inquisitiveness" , "Inquisitive characteristic" , "Inquisitive quality" , "Curiosity of nasal kind" .)