I found John H's marriage to Mary Kenny on FreeBMD (limited search to all marriages in the uk between John Baugh's and Mary's (only 2 in the West Midlands - one in Wolverhampton who was a Mary Jane anyway, and the correct one in Walsall in December 1881). It also gives his middle name as Henry. I have also found the record on Ancestry (limited search to Walsall this time and found all marriages of John Baughs between 1870-1890 and none of the other records - 5 in total - matched).
I then tried to look for a census return in 1871 in walsall for a Mary Kenny who was born c1863 in Walsall. There was only one (born c1864). Her father Thomas was a labourer from Galway. Mary Ann was 7 on the 1871 census, but I have traced the family on the 1861 census also but no further than that. It's likely that they came over in the 1840s/1850s, along with many others fleeing the potato famine. I had wondered if there was an Irish connection somewhere, as Maureen (my partner John's mum) had said previously that her grandmother Annie Baugh (later Simmons) had spoken of one, although she had thought that it related to Annie herself. I am happy to send you copies of the census returns. I can send a copy of the marriage record too, although it doesn't give much information - I haven't ordered the certificate yet.
I will now try to send you the message I prepared earlier!
Jenny