University angered cooks lacking the required seniority (8)
I believe the answer is:
underage
'lacking the required seniority' is the definition.
'university angered cooks' is the wordplay.
'university' becomes 'u' (**).
'cooks' indicates an anagram.
'angered' with letters rearranged gives 'nderage'.
'u'+'nderage'='UNDERAGE'
(Other definitions for underage that I've seen before include "Not legally old enough" , "Too tender" , "one too young" , "Not yet eligible to vote - too young" , "Not old enough to exercise legal right" .)