Give up extremely detestable sort of meat (7)
I believe the answer is:
Here is my best explanation:
'give up' is the definition.
('deliver' can be a synonym of 'give up')
'extremely detestable sort of meat' is the subsidiary indication.
'extremely' means to remove the middle letters.
'sort of meat' becomes 'liver' (**).
'detestable' with its centre removed is 'de'.
(Other definitions for deliver that I've seen before include "Rescue", "Dispatch", "Formally hand over", "Transfer and hand over", "Give out", "Bring letter or rescue", "Bring to a destination and hand over", "Bring to its destination - rescue", "The highwayman's cry - 'stand and . . . . . . .'" and "Hand over (letter)".)