Deeply, oddly enlightened state (8)
I believe the answer is:
delaware
'state' is the definition.
(I know that Delaware is a state)
'deeply oddly enlightened' is the wordplay.
'oddly' indicates alternate letters (letters in odd-numbered positions).
'enlightened' becomes 'aware' (I've seen this in another clue).
The alternating letters of 'deeply' are 'del'.
'del'+'aware'='DELAWARE'
(Other definitions for delaware that I've seen before include "American state, capital Dover" , "State of northeast US" , "state of the east" , "Second smallest US state" , "State of eastern US" .)