Fight cheers Greek city state (6)
I believe the answer is:
sparta
'city state' is the definition.
(I know that Sparta is or was a city state)
'fight cheers' is the wordplay.
'fight' becomes 'spar' (sparring is a kind of fighting).
'cheers' becomes 'ta' (both can informally mean thank you).
'spar'+'ta'='SPARTA'
'greek' acts as a link.
I am not very happy about this link. It may belong to another bit of the clue.
Can you help me to learn more?
(Other definitions for sparta that I've seen before include "Ancient Greek city noted for its austerity" , "Ancient country" , "Ancient Greek city where great self-denial was practised" , "in ancient Greece" , "Ancient Greek city whose citizens were famed for toughness" .)