Give up extremely detestable sort of meat (7)
I believe the answer is:
deliver
'give up' is the definition.
('deliver' can be a synonym of 'give up')
'extremely detestable sort of meat' is the wordplay.
'extremely' means to remove the middle letters.
'sort of meat' becomes 'liver' (**).
'detestable' with its middle taken out is 'de'.
'de'+'liver'='DELIVER'
(Other definitions for deliver that I've seen before include "Convey, transport (goods)" , "Render" , "bear" , "Convey to destination" , "Formally hand over" .)