The couple are together 1921 at the address on DC, her surname Harmer. Joseph there from around 1919 on RN service records.
I think the note on DC could be as his wife Martha claimed his RN pension rights. Maybe she contested that M.E Brooks, who registered death, as widow, was M.E Harmer and not his wife?
Just a suggestion, I would consider DNA testing, by a elder family member, to try confirm parentage of Jack/Joseph jnr. DNA cannot lie.
Cas