Irregularly, stayed outside university for part of the week (7)
I believe the answer is:
tuesday
'part of the week' is the definition.
(I've seen this before)
'irregularly stayed outside university' is the wordplay.
'irregularly' is an anagram indicator (I've seen this in other clues).
'outside' is an insertion indicator.
'university' becomes 'u'.
'stayed' with letters rearranged gives 'tesday'.
'tesday' going around 'u' is 'TUESDAY'.
'for' acts as a link.
(Other definitions for tuesday that I've seen before include "Twenty-four hours" , "Part of week (for pancakes?)" , "Day of the week" , "Day the Romans called the day of Mars" , "Is it suet day?" .)