Looking for John Cunningham born c1907

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'By elimination other person is Mary, born 1934'

Using free search for Mary Cunningham, 1934, Leeds, shows her as Cunninghan/Flanagan, and as she is showing she has died.  You can get a free search but not see the actual copy of the register through FindMyPast 1939 register.

https://www.findmypast.co.uk/1939register

Look then for a marriage and any children (sorry, not on usual computer etc at the moment) for Mary.

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The Louisa showing in 1939 might be his second wife, because there is this possible marriage:

Marriages March quarter  1939   
 John Cunningham   Leeds North   
 Louisa Lowther    Leeds North   

There is also the possibility that Louisa might have been married before, so the blacked out entry could be a child of a first marriage for her, or a totally unrelated person.

Mary is shown as Cunningham/Flanagan, and I have previously posted a possible marriage.

This is followed by a death
Deaths June quarter 1960 
Mary C Flanagan age 25 in Leeds   2c   144   

Cannot see any births for children of the marriage.

There is this possible first marriage for John:
Marriages June quarter 1933 
John W Cunningham    Leeds North   
Clara Howden     Leeds North   
but cannot see a death for Clara (nor a Leeds birth for Clara Howden)

It depends how much you want to spend to try and check if these are the correct records.