Think this maybe the 1871 for Ann Byrne. Age is out but image could be 31. So likely unmarried on marriage to McAndry? I would not assume relationship until more evidence. Connection could be a Burn/Byrne cousin...Jane could have remarried a Welsh etc.
Godparents are normally sponsored to step in if anything happens to parents if I am correct. I think there are clues in the children's bapts as godparent relationships can be family or close friends
1871 - All Souls, Manchester (New Cross)
Anne Byrne 21 Head, unmarried, Ireland
Jane Welsh 56 Mother, Widow, Ireland
Anne Mee 15 Boarder
Mary Hunt 70 Boarder
Anne Hunt 28 Boarder
If access -
https://www.ancestry.co.uk/search/collections/7619/records/17718563I would try use DNA. Have you tried a search for surnames and area in your DNA tab to see if on right track with any hits? Can be tricky finding connection, but the search in DNA tab all have some link to you.
Added - would remember there is over 4 yrs between 1891 census and marriage address in 1895, would check the surname of people/wife at marriage address for possible family connection maybe.