I was able to transcribe this much from a sound sample:

...I was born at the daffodil time of the year
So they just called me Daffy, now maybe that's queer.
But I'm—

[CHORUS?] Sweet Daffodil Mulligan I am
All my ancestors come from The Coombe
And I'm just such another, the spit of my mother,
The lady that lives in one room (Down in Francis Street)
Sweet Daffodil Mulligan (Fresh fish!)...

Google Book Search found this review in "The Gramophone" (1939) edited by Compton Mackenzie and Christopher Stone:

Humour, such as Jimmy O'Dea (1899-1965) and Harry O'Donovan give in Fresh Fish is hard to appreciate when played "cold" to a microphone. A duologue such as this requires atmosphere, and apart from a few Irishisms there is none on this side of Regal-Zonophone MR3109. Still, O'Dea makes some amends for the dullness of this side on the reverse in another example of Irish humour, a song entitled Sweet Daffodil Mulligan....