From searches a couple of weeks ago, this is the only entry I could find for a John White born in Fulham c. 1893. Entry in Chelsea in 1901:
John W White 65 b. Hutton Essex
Maria White 64 b. Hutton Essex
Annie W White 32 dressmaker b. Saint J, Westminster
Lily I White 19 dressmaker b. Chelsea
John White 8 Grandson b. Fulham
Family Search have only partial entry
www.familysearch.org/ark:/61903/1:1:X9HH-VFMHowever, I couldn't see an Edmund Joseph connected to this family . Going back over the census with them, the only sons I could see were:
1881
www.familysearch.org/ark:/61903/1:1:Q273-H4QK1891
www.familysearch.org/ark:/61903/1:1:QQPB-8ZM Also showing on original image is Sydney aged 16 and Lilley S age 8. Not sure why not on FS

Maria White is widowed by 1911
www.familysearch.org/ark:/61903/1:1:XWLZ-Y1D Grandson John White not showing with them.
Let's see what comes back on that birth entry

Monica