Diagnose rash in US city (3,5)
I believe the answer is:
san diego
'us city' is the definition.
(San Diego is a city in California)
'diagnose rash' is the wordplay.
'rash' is an anagram indicator (a reckless or careless ordering of the letters).
'diagnose' anagrammed gives 'SAN DIEGO'.
'in' is the link.
(Other definitions for san diego that I've seen before include "Pacific port" , "Californian resort" , "Agonised over the Californian city" , "Southern California naval base" , "California naval port" .)