Angry, promise to produce puzzle (9)
I believe the answer is:
crossword
'puzzle' is the definition.
(type of puzzle)
'angry promise' is the wordplay.
'angry' becomes 'cross' (cross can mean annoyed or angry).
'promise' becomes 'word' (as in 'I give you my word').
'cross'+'word'='CROSSWORD'
'to produce' acts as a link.
(Other definitions for crossword that I've seen before include "Puzzle like this one" , "Letter puzzle" , "Puzzle (for the intelligent)" , "Puzzle with squares" , "See 5" .)