Crazy child conceals his second race (7)

I believe the answer is:

madison

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'race' is the definition.
Although both the answer and definition are singular nouns, I cannot see how they can define each other.

'crazy child conceals his second' is the wordplay.
'crazy' becomes 'mad' (synonyms).
'child' becomes 'son' (male child).
'conceals' indicates putting letters inside.
'his second' becomes 'i' (2nd letter of 'his').
'mad'+'son'='madson'
'madson' placed around 'i' is 'MADISON'.

Can you help me to learn more?

(Other definitions for madison that I've seen before include "James -, fourth president of the USA" , "US President - county with bridges" , "-- Avenue, New York's advertising district" , "New York Avenue for advertising people" , "US President and Avenue" .)

I've seen this clue in The Daily Express.

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