Yes, I'm still looking!
Here's what a friend produced for me:
I searched for Barkham's in Dudley and had one good hit. A Trevor
Onions, born 1941, in Dudley, Mothers maiden name Barkham. There were
no more entries for new born Onion's with the mothers maiden name
being Barkham.Thinking there may be a spelling error, i discovered a year or two
later another newborn Onion, had a mother who's maiden name was Barklam.
A check for Barklam shows a family, in Handsworth, Yorkshire, of John
and Annie Barklam, both born in Dudley! Their children were born in,
Yorkshire. One child, a George A. Barklam, born 1891, died in the third quarter of 1915, in the district of Sheffield!.
George A. Barklam has no CWGC entry. I've searched for other possibilities, but with no results. The family were in Yorkshire in 1883, 30 years before the war. Perhaps a relative in the 1920's recalled a death in their family but could not recall the name, only
the incorrect surname.
So the hunt goes on...