Fight cheers Greek city state (6)
I believe the answer is:
'greek city state' is the definition.
'sparta' can be an answer for 'state' (I have seen 'Laconic state' mean 'sparta' so perhaps 'state' could also mean 'sparta'). I'm unsure of the 'greek city' bit.
'fight cheers' is the wordplay.
'fight' becomes 'spar' (sparring is a kind of fighting).
'cheers' becomes 'ta' (both can informally mean thank you).
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" .)