Thanks Pels and Gadget. I had considered the possibility that Jessie was Mary's child, which would explain her being called Jessie Garnett, daughter, in 1901. Didn't know how to go about research over the border. I'm amazed at your detective skills! I only just found out that Jessie, Janie, Jinny and Jenny were pet names for Janet (and also for Jean in Scotland).
Just quizzed an elderly relative who tells me he knew Mary was born in Scotland but she didn't have a scottish accent. Also, maybe I should have added that on Mary and Henry Garnett's marriage certificate, Mary is recorded as Mary Rafter, widow (formerly Barrow). I can't find a Jessie Rafter either. This made me wonder if Mary claimed to be a widow in order to legitimise her daughter.
If the 1871 census is the right one are we assuming then that Mary was Ellen's illegitimate daughter? In 1891 Salford James and Ellen have nine children listed, three of them born in Salford before that marriage in 1880.
In 1881 James Barrows birthplace is given as Exeter, Devon and 1891 it's Salford.
Lyn