State without a health centre (4)
I believe the answer is:
utah
'state' is the definition.
(Utah is a US state)
'without a health centre' is the wordplay.
'centre' says the answer is hidden in the clue (hidden in the middle of the letters).
'UTAH' can be found hidden inside 'without a health'.
(Other definitions for utah that I've seen before include "State of Western USA" , "State of the union" , "American state, capital Salt Lake City" , "The American Mormon state" , "A hut for the Mormons, oddly" .)