I think you may have a few details wrong. John PAYNE & Anna HUNTER married in 1893 which would mean that Sydney was born 27 years after they married - not impossible but not likely.
I checked out the Ellis Island passenger lists.
There is a Sydney PAYNE age: 3 travelled from Southampton to New York 1923 Ship: Olympic.
It states that his las residence was Hardingham Norfolk but his place of birth was Nottinghamshire. His mother paid his passage and he was going to Brotherinlaw Albert GASCON (U.S.C) Box 85 Sea Breeze Rochester N.Y.
A Violet GASCON age: 24 wife of Albert GASCON same address was also on the ship. Now there is a birth in UK that fits
Births Qrt Dec 1919
PAYNE Sydney H L Mother's Surname: PAYNE
District: Nottingham Vol: 7b Page: 890
My guess would be that Sydney's mother may have been unmarried at time of his birth. Is it possible that his mother was Violet GASCON (nee Payne)...need to see if I can find a marriage of a Violet Payne to an Albert GASCON.
Found 1930 USA Census
Sidney PAYNE b: 1920 is with Albert & Violet GASCON on the 1930 Census. But he isn't on the 1940 Census with the family. Which now has 5 sons and two daugthers. 1940 Census has mother born England and they are in 18 Elser Ter Rochester , Monroe NY.
I will try and find marriage of Violet ? and Albert GASCON.