There are four Castles of Dromore in Ireland located in the counties of Down, Kerry, Limerick and Tyrone. No one can say for certain which Castle Dromore the song is about.

Castle of Dromore - Irish Children's Songs - Ireland - Mama Lisa's World: Children's Songs and Rhymes from Around the World  - Intro Image

Notes

(1)The banshee is "a female spirit in Irish mythology who heralds the death of a family member, usually by wailing, shrieking, or keening." (Wikipedia) This verse seems to be about the mother's hope that her baby doesn't die in infancy.
(2) Or "rose".
(3) Or "our land".

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Tune: Old Irish Air (possibly from the song "My Wife is Sick").
Lyrics: Harold Boulton (1892). Boulton was from England.

There's an Irish Gaelic translation by Douglas Hyde that was written later.
Link to Score with English and Gaelic lyrics.

Castle of Dromore - Irish Children's Songs - Ireland - Mama Lisa's World: Children's Songs and Rhymes from Around the World  - Comment After Song Image
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Thanks and Acknowledgements

Photo: Mike Searle / Castles of Munster: Dromore, Clare / CC BY-SA 2.0.

Image of Banshee: Bunworth Banshee, "Fairy Legends and Traditions of the South of Ireland", by Thomas Crofton Croker, 1825.