Baseball player heading off to go round American Western state (4)
I believe the answer is:
utah
'western state' is the definition.
(western US state)
'baseball player heading off to go round american' is the wordplay.
'baseball player' becomes 'Ruth' (Ruth is an example).
'heading off' means to remove the first letter.
'to go round' is an insertion indicator.
'american' becomes 'a' (common abbreviation - e.g. in organisation names).
'ruth' with its initial letter taken off is 'uth'.
'uth' enclosing 'a' is 'UTAH'.
(Other definitions for utah that I've seen before include "Mormons are found here in a hut" , "Bryce Canyon US state" , "America's Mormon state" , "Mormon US state" , "State of the union" .)