Changing the subject, that was a great fight (extremely late starting) (12)

'extremely late starting' is the definition.
The definition suggests a singular noun which matches the answer.

'changing the subject that was a great fight' is the wordplay.
I cannot quite understand how this works, but
'the' could be 't' (the is pronounced as a 't' sound in some dialects) and 't' is located in the answer.
'a' is found in the answer.
'fight' could be 'bout' (as in a bout in boxing) and 'bout' is present in the answer.
This may be the basis of the clue (or it may be nonsense).

Can you help me to learn more?