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In 1901, William Smith, 27 yrs states he is born Sheffield.
This looks like the family in 1911 where William Smith shows born Staffordshire 
https://www.familysearch.org/ark:/61903/1:1:XW13-9W4

In 1921, however, there is William Smith, b 1872 , Suffolk with wife and children looking very like the 1911 family.

This looks to be the marriage
1896 Salford Register Office or Registrar attended
William Smith and Elizabeth Wolfe

Looks like I owe you an apology. Having looked at the 1911 William Smith census entry you mentioned, I can see now that it is indeed the same person as the 1901 census. Everything is identical apart from the place of birth. And the two properties are 100 m apart and another 100m to where their mother was living in 1911. Why did he say he was born in Staffordshire? Did someone else fill in the census form and got it wrong? I did manage to find Elizabeth Wolfe's birth entry on the Scotland's People site. She is a Glaswegian like myself. 

1921 is only important at the moment because of the Suffolk birthplace for William really.

I had a quick look round earlier and 1901/1911 William and family and then the sighting of Mary Ann are in close proximity.
It is the mystery of Mary Anne’s whereabouts in 1901 and no death for Joseph that are problematic.
If they had fallen on hard times, you would think Mary Ann would be with her children.
Perhaps the couple separated.

I think they must have separated. The children would all have left home by then. I do have Joseph in a Manchester workhouse in July 1905 (GBPRS_MAN_4509706_00014) so obviously fallen on hard times. I still think this:-
[Edit.... image removed]

Is Mary Ann Gavin in 1911. The stated birthplace is 12 miles from she was actually born but all other details match her.

Quote from: Vendee on Wednesday 28 December 22 18:36 GMT (UK)

Quote from: heywood on Wednesday 28 December 22 16:16 GMT (UK)

I think she is Theresa Gavin, b Sheffield at 1901 3731/59/15 boarding with William Smith b Sheffield and family in Salford.  Would that be her half brother?

If there is a William Smith on that same entry then I'm not sure it is her half brother. He was born in Suffolk and would have been 28 in 1901. Last thing I heard about William was on the 1891 census when he  an "inmate" at the Market Weighton Roman Catholic reformatory in Yorkshire, have been sent there for 5 years in 1897. I didn't think he ever joined up with his family in Salford but I may be wrong.

In 1901, William Smith, 27 yrs states he is born Sheffield.
This looks like the family in 1911 where William Smith shows born Staffordshire 
https://www.familysearch.org/ark:/61903/1:1:XW13-9W4

In 1921, however, there is William Smith, b 1872 , Suffolk with wife and children looking very like the 1911 family.

This looks to be the marriage
1896 Salford Register Office or Registrar attended
William Smith and Elizabeth Wolfe

Thanks again for your help. I'm trying to pick through the information I've been getting here. Your link to the Staffordshire William Smith isn't my relative. He was definitely born/registered in Kedington, Suffolk. The other thing is that once his mother Mary Ann Smith moved to Sheffield and married Joseph Gavin, William appears in the 1881 census as William Smith but in the 1891 census during his 5 year spell at the reformatory school he was William Gavin. The press cuttings and reformatory records from his stealing offence also refer to him as William Gavin. I'm not sure if he would have reverted to Smith again but as I mentioned earlier in the thread, family members were not averse to changing surnames when it suited them.

Mary T Gavin married Patrick Hoben at Salford St Simon. I can’t  online records for that church.

I think she is Theresa Gavin, b Sheffield at 1901 3731/59/15 boarding with William Smith b Sheffield and family in Salford.  Would that be her half brother?

Ah... right. I can't find my copy of Mary Theresa's 1901 census on my PC at the moment but my notes say that her status on that entry was "boarder". If there is a William Smith on that same entry then I'm not sure it is her half brother. He was born in Suffolk and would have been 28 in 1901. Last thing I heard about William was on the 1891 census when he  an "inmate" at the Market Weighton Roman Catholic reformatory in Yorkshire, have been sent there for 5 years in 1897. I didn't think he ever joined up with his family in Salford but I may be wrong.

Quote from: Vendee on Wednesday 28 December 22 11:30 GMT (UK)

Quote from: heywood on Tuesday 27 December 22 21:58 GMT (UK)

This may be Joseph in 1901 3719/37/13

Joseph Gavin, 62 yrs Navvy b Ireland lodging in Pendleton

Yes I've got that record thank you. Makes you wonder why there is no 1901 record for Mary Ann or why they were not living together although he still declares himself as married. I do accept that there are some things that I'll never find out.

The census is only a snapshot of one night in ten years. It’s possible that Joseph was simply working away from home for a time.

Added: a guess - U for Union?

Thanks. I agree that the census is a snapshot but Joseph had been a Salford resident for over 10 years so he wasn't away from home in 1901 (Pendleton is in Salford)