https://www.familysearch.org/ark:/61903/1:1:NTHR-D6V - according to Hilda's baptism she was Bertha Eliza
Wright's father was a John, as was Bertha's, so perhaps "John" was just taken from one of Clarence's grandfathers?
I think Bertha went on to marry Harry Jackson in Bramley district in early 1914, and had a son later that same year, Norman (mother's maiden name "Bromwick"), who sadly died as an infant.
Any indication of the occupation of "John" on the baptism?
Was he born at Lockwood? Was that also the address of the informant (possibly Lily) on the birth certificate? She may have only been at the workhouse to give birth, not being able to afford medical assistance otherwise.