Fight, thanks to old city state (6)
I believe the answer is:
sparta
'old 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 'old city' bit.
'fight thanks' is the wordplay.
'fight' becomes 'spar' (sparring is a kind of fighting).
'thanks' becomes 'ta' (UK informal version of 'thank you').
'spar'+'ta'='SPARTA'
'to' acts as a link.
Can you help me to learn more?
(Other definitions for sparta that I've seen before include "Ancient Greek city-state" , "Ancient Greek city state" , "Formerly "City" , "Old Greek city state, famous for its citizens' austere lifestyle" , "A strap from an ancient Greek city?" .)