This is one of those songs which has so many variants that it is hard to tell what is just a different version of a particular song, or what is a fragment of it, or what represents a 'different ' song altogether.
The song was widespread in the 19th century, but it took on a lot of local "colourings", as can be seen by the number of ballad sheets just in the Bodleian alone.
"The Irish girl" and "The New Irish Girl"

1) Ballad sheet: printed by W. McCall, Liverpool, between 1857 & 1877
The Irish Girl (McCall #1)
The Irish Girl (McCall #2)

2) Ballad sheet: printed by T.Hogget, Durham, between 1816 & 1843
The Irish Girl (Hogget #1)
The Irish Girl (Hogget #2)

3) Ballad sheet: printed by Wm. Armstrong, Liverpool, between 1820 & 1824
The Irish Girl (Armstrong #1)
The Irish Girl (Armstrong #2)

4) Ballad sheet: printed by W. Stephenson, Gateshead, between 1821 & 1838
The Irish Girl (Stephenson)

5)Ballad sheet: printed by J.Evans, London, between 1780 & 1812
The Irish Girl (Evans)

6) Ballad sheet: printed by J.Catnach and others, London, between 1813 & 1838
The Irish Girl (Catnach)

7) Ballad sheet: no imprint, no date, (only partially legible)
The Irish Girl (n.a.)

8) Ballad sheet: printed by J. Harkness, Preston, between 1840 & 1868
The Irish Girl (Harkness)

9) Ballad sheet: no imprint, no date
The Irish Girl (undated)

10) Ballad sheet: printed by J.Lindsay, Glasgow, between 1851 & 1910
The Irish Girl (Lindsay)

11) Ballad sheet: printed by W. Carse, Glasgow, c. 1825
The Irish Girl (Carse

12) Ballad sheet: Printed by T. Batchelar, London, between 1817 & 1828
The Irish Girl (Batchelar)

13) Ballad sheet; Utterly illegible (!)
The Irish Girl (Illegible)

14) Ballad sheet: Printed by H. Such, London, 1849 - 1864
The New Irish Girl (Such #1)
The New Irish Girl (Such #2)

15) Ballad sheet; Printed by J.Pitts, London, between 1819 & 1844
The New Irish Girl (Pitts #1)
The New Irish Girl (Pitts #2)
The New Irish Girl (Pitts #3)
The New Irish Girl (Pitts #4)
The New Irish Girl (Pitts #5)