Sick Shakespeare's animosity? (3,4)
I believe the answer is:
ill will
'animosity?' is the definition.
(animosity is a kind of ill will)
'sick shakespeare's' is the wordplay.
'sick' becomes 'ill' (synonyms).
'shakespeare' becomes 'will' (I've seen this in other clues).
'ill'+'will'='ILL-WILL'
(Other definitions for ill will that I've seen before include "Disposition of enemy" , "Hostility, malevolence" , "Bitterness" , "Positively unfriendly disposition" , "Animosity, hostile feeling" .)