Terrible anger after final demand (4)
I believe the answer is:
dire
'terrible' is the definition.
('dire' can be a synonym of 'terrible')
'anger after final demand' is the wordplay.
'anger' becomes 'ire' ('ire' can be a synonym of 'anger'**).
'after' means one lot of letters go next to another.
'final' indicates one should take the final letters.
The last letter of 'demand' is 'd'.
'ire' after 'd' is 'DIRE'.
(Other definitions for dire that I've seen before include "Nearly hopeless" , "Of very poor quality" , "Ride in dreadful state" , "to speak (13) [FRENCH]" , "Ominous; very bad" .)