Animosity expressed by ailing Bard (3,4)
I believe the answer is:
ill will
'animosity' is the definition.
(I've seen this before)
'ailing bard' is the wordplay.
'ailing' becomes 'ill' (similar in meaning).
'bard' becomes 'will' (William Shakespeare).
'ill'+'will'='ILL-WILL'
'expressed by' is the link.
(Other definitions for ill will that I've seen before include "Positively unfriendly disposition" , "Hostile, unfriendly feeling" , "Hostility, malevolence" , "Hostility, unkind feeling" , "Bitterness" .)