I'm late joining this thread and hope I've followed it all. You are looking for the birth of John Henry Wood circa 1873 in Dewsbury reg district. On census returns where he's been identified, his year of birth is consistent at circa 1873, and place of birth Heckmondwike. I have searched the original GRO births qrt x qrt 1871 - 1873 and didn't find anything suitable. There are six John Wood's born 1871-1875 in Dewsbury, but no John Henry. Of course he may have been registered as John Wood and Henry added at his baptism. The John Hey Wood birth Dob's found could be him, the registrar may have entered him as John Hry Wood and it was copied wrongly, though admittedly that birth seems a little late, however it wouldn't be the first time someone has gone through life believing themselves to be older than they actually are.
Within the Wood family there's definitely a Heckmondwike area connection. In 1871 George and Kate, children of James are living in Heckmondwike. RG10/Piece4590/Folio75/Page46.
They are still in Heckmondwike in 1881. RG11/Piece 4553/Folio 107/Page 27.
And this could be James in 1871, his age is out by a few years but all other details fit, he's an iron moulder, a widow, and born Rotherham. RG10/Piece4564/Folio140/Page29.
Apologies if I've repeated anything previously posted. My info doesn't take us nearer to John Henry's birth, but does suggest a Heckmondwike connection.
Is the marriage between James Wood and Hannah Sykes a definite, or a maybe? I've searched census returns but cannot find them together. Do we know when James died?