The horror of one’s experience under duke (8)
I believe the answer is:
distaste
'the horror' is the definition.
(I have seen 'Mild horror' mean 'distaste' so perhaps 'horror' could also mean 'distaste')
'one's experience under duke' is the wordplay.
'one's' becomes 'is' (Roman numeral).
'experience' becomes 'taste' (taste is a kind of experience**).
'under' says to put letters next to each other.
'duke' becomes 'd'.
'is'+'taste'='istaste'
'istaste' put after 'd' is 'DISTASTE'.
'of' is the link.
(Other definitions for distaste that I've seen before include "Adverse reaction" , "Mild horror" , "mild revulsion" , "Loathing" , "Antipathy, aversion" .)