Herb makes duke sick (4)
I believe the answer is:
dill
'herb' is the definition.
(dill is a kind of herb)
'duke sick' is the wordplay.
'duke' becomes 'd'.
'sick' becomes 'ill' ('ill' can be a synonym of 'sick').
'd'+'ill'='DILL'
'makes' is the link.
(Other definitions for dill that I've seen before include "Herb; rhymes with beak" , "Herb commonly used with gravlax" , "Herb used in cooking" , "Herb for gravadlax" , "Herb used in pickles" .)