Aristocrat from this country cutting half of debt (4)
I believe the answer is:
duke
'aristocrat' is the definition.
(duke is a kind of aristocrat)
'this country cutting half of debt' is the wordplay.
'this country' becomes 'uk' (United Kingdom).
'cutting' indicates putting letters inside (some letters cut their way into a word).
'half of' means to take half the letters.
'debt' cut in half is 'de'.
'uk' put within 'de' is 'DUKE'.
'from' acts as a link.
(Other definitions for duke that I've seen before include "Highest ranked British peer" , "Highest ranking British peer" , "- - Ellington (jazz musician)" , "Member of the nobility" , "Highest-ranking nobleman" .)