Period of not eating quickly (4)
I believe the answer is:
fast
I believe this clue is a double definition.
'period of not eating' is the first definition.
'quickly' is the second definition.
('fast' can be a synonym of 'quickly')
(Other definitions for fast that I've seen before include "Period during which fare is reduced" , "Go without food" , "Not eating food" , "Don't scoff" , "Eat nothing" .)