I am leaning to the poor house record in my original post, however it is a poor House record only and has no parents listed.
What information does it contain? Does it name a husband, or a birthplace, or say how many children she had? Why was she admitted to the Poorhouse?
There's a J C, female, married, 38, born Kilbrandon, Argyll, in the 1861 census, but she is a patient in a private lunatic asylum so she is probably not the one who was admitted to Barnhill in 1854 and died in 1861. Also there's no way of knowing if the C stands for Cumming. That Springburn death in 1861 will be the correct one for the Barnhill inmate so it looks unlikely that she is the wife of Peter Cumming.
I reckon, for what it's worth, that Jean Martin, wife of Peter Cumming, died between the 1851 census and the start of civil registration in 1855. Peter left his sons in the care of his mother-in-law and went to Australia, where he got himself established and then sent for the boys to join him. That would explain the apparent absence of both Peter and Jean from the 1861 census.
I am not a member of Scotland’s people
No-one is a 'member' of Scotland's People. It's not a membership or a subscription service. It is unavoidable if you are researching family in Scotland.
See
https://www.rootschat.com/forum/index.php?topic=714261.0