Good health initially found in a supporter’s dog (6)
I believe the answer is:
afghan
'dog' is the definition.
(Afghan hound is a breed of dog)
'good health initially found in a supporter's' is the wordplay.
'initially' suggests taking the first letters.
'found in' is an insertion indicator.
'supporter' becomes 'fan' (fan is a kind of supporter).
The first letters of 'good health' is 'gh'.
'a'+'fan'='afan'
'gh' put within 'afan' is 'AFGHAN'.
(Other definitions for afghan that I've seen before include "tongue" , "Knitted blanket - sheepskin coat - large dog" , "from somewhere in central Asia" , "Type of coat made from sheepskin" , "Native of an Asian republic" .)