Terrible greed that consumes an important person (7)
I believe the answer is:
grandee
'important person' is the definition.
'terrible greed that consumes an' is the wordplay.
'terrible' is an anagram indicator.
'that consumes' means one lot of letters goes inside another.
'greed' anagrammed gives 'grdee'.
'grdee' enclosing 'an' is 'GRANDEE'.
(Other definitions for grandee that I've seen before include "One of high rank" , "Enraged (anag) - Spanish nobleman" , "Eminent man" , "Spanish bigwig" , "Angered over Spanish noble" .)