Discover the absence of our island state (8)
I believe the answer is:
missouri
'state' is the definition.
(Missouri is an example)
'discover the absence of our island' is the wordplay.
I cannot quite see how this works, but
'of' could be 'o' and 'o' is located in the answer.
'island' could be 'is' (abbreviation) and 'is' is present in the answer.
This explanation may well be incorrect...
(Other definitions for missouri that I've seen before include "American state's" , "So is rum I alter to the state" , "24th American state to join the Union, 1821" , "Biggest tributary of the Mississippi" , "Flower" .)