Got sick with American relish (5)
I believe the answer is:
gusto
'relish' is the definition.
(I know that relish can be written as gusto)
'got sick with american' is the wordplay.
'sick' is an anagram indicator.
'with' indicates putting letters inside.
'american' becomes 'us' (abbreviation for United States).
'got' anagrammed gives 'gto'.
'gto' going around 'us' is 'GUSTO'.
(Other definitions for gusto that I've seen before include "Vigorous and enthusiastic enjoyment" , "Energy" , "Relish; vigour" , "Great enjoyment, vigour" , "Obvious enthusiasm" .)