Thank you so much for your reply. The spelling of McGurk was found in a will and on a death certificate too. But McGuirk does seem the most likely spelling.
Because Elizabeth's boys were all born Sligo i am only guessing she was born there along with her sister. Both Elizabeth's and Bridget's husbands were both born County Tyrone. So they could have been born there, who knows.
I have searched for Elizabeth's birth in Sligo but had no results.
Thank you so much for the links they will be of great use.
Thanks again for your help, it's most appreciated.