Here follows the four lines of the song cintaining the reference and the totally incorrect interpretation of "asthore".
As others have already said, it means "my dear / darling / sweet etc."

"There's a letter I'll be mailing for soon I will be sailing,
And I'll bless the ship that takes me to my dear old Erin's shore.
There I'll settle down forever, I'll leave the old sod never,
And I'll whisper to my sweetheart, "Come and take my name asthore."* (*asthore: Irish for "forever.")

http://cjtm.icaap.org/
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Annraoi

And the URL for the Canadian Journal for Traditional Music is: