St Rollex - she died in Glasgow Poorhouse, but it is a good bit after George died, and he said he was a widower.
If she had deserted him, perhaps he put it about that she had died to avoid the stigma of having been deserted? Who registered his death?
From the IGI the dates of birth of the children in Scotland are
Catherine 13 May 1857
Miles 2 August 1859
John 9 August 1861
James 18 January 1863
Martha 18 May 1865
Ann 15 July 1867
all in Greenock
The ages in the 1861 census suggest that Edward was born 1848/1849 in Ireland, and Patrick 1853/1854 in Greenock.
The RC baptisms on Scotland's People all list the mother as Mary Murray not McMurray
Ann born 27 Oct 1851 bap 2 Nov 1851 St Michael's Lennoxtown
Pat born 7 Jun 1853 bap 12 June 1853 St Mary's Greenock
Ann born 23 Apr 1855 bap 17 June 1855 St Mary's Greenock
Catherine born 13 Mar 1857 bap 17 May 1857 St Mary's Greenock
Nicolaus born 2 Aug 1859 bap 8 Aug 1859 St Mary's Greenock
Curious that there is no birth listing for Ann b 1855 in SP. Also unfortunate because if the birth had been registered the certificate should have contained lots of useful extra infrmation.
As Mary died in the poorhouse, it would be worth checking to see if there is any information about her in the Glasgow parochial board (Poor Law) records in the Mitchell Library.