Brief period that may produce famine (8)
I believe the answer is:
shortage
'famine' is the definition.
(synonyms)
'brief period' is the wordplay.
'brief' becomes 'short' (synonyms).
'period' becomes 'age' (age is a kind of period).
'short'+'age'='SHORTAGE'
'that may produce' is the link.
(Other definitions for shortage that I've seen before include "Want" , "Deficit" , "Horse tag (anag.)" , "Lack, deficiency" , "Insufficiency" .)