So after a lot of digging it turns out Margaret was the daughter of Thomas Legh of Alkrington (son of Piers and Margaret Legh of Lyme) and his wife Katherine.
Thomas's 1597 will names 7 'reputed' children (including Margaret Legh) and that it was his will and intent that his wife Katherine had 'education, rule, order, government and disposition' of his 3 reputed daughters and 3 reputed sons, the eldest already being an adult with his own children.
Katherine's 1620 will names them all as her sons and daughters and she mentions 'my daughter Starkie' and 'sonne in lawe John Starkie' as her 'true and lawfull executor of this my last will and testamt'.
So it appears that the children were illegitimate even though sources say Thomas and Katherine married in the early 1560s, with Margaret being born in the mid-late 1580s. For some reason their marriage must have been invalid, rather than Thomas having had 7 children with a different woman than his wife Katherine.
Thomas and Katherine appear to share the same great-grandparents Edmund Trafford and Margaret Savage, making them second cousins. I wonder if this was reason for making issue illegitimate? I remember seeing a while ago a couple divorced in the late 14th century because they were related in the 'fourth degree', both descending from the same great-great-grandparents. Maybe somebody can shed some light on this?