State university — teach with some regularly absent (4)
I believe the answer is:
utah
'state' is the definition.
(Utah is a US state)
'university teach with some regularly absent' is the wordplay.
'university' becomes 'u' (abbreviation for university).
'with some regularly absent' means one should take alternating letters (letters removed at regular intervals).
The alternate letters of 'teach' are 'tah'.
'u'+'tah'='UTAH'
(Other definitions for utah that I've seen before include "US state, capital Salt Lake City" , "US state N. of Arizona" , "State of western US" , "Brigham Young's state" , "In USA, it's the Mormon state" .)