On another thread called Loving Hannah, it was Martin Ryan (thanks, Martin) who identified Davy Hammond for me as the singer on a cassette of "Irish Folk Songs". On the cassette, he starts with "B"is for Barney and then ends on side two with "Fan-a-Winnow", inserting some of the B is for Barney lyrics into Fan-a-Winnow.
Here is a cut and paste from that thread:

Steve, I found the recording of the Irish Girl, with the slow sad tune similar to Circle be Unbroken. I don't know if you will be able to find this tape... but it's a good one. It is of anonymous singers, and the song is sung by an old Irishman. It is called simply, "Irish Folk Songs" and was in the bargain rack for about $6.US. It is made in Candada, the cassette has written on it, Excelsior International, Distributed by the Madacy Music Group, Inc, P.O. Box 1445, St-Laurent Quebec, Canada H4L 4Z1.

Songs listed are,
B is for Barney,
All Around the Loney-O,
Green Gravel,
The Doffin Mistress,
The Maid of Ballydoo,
I Know My Love, vThe Dark Eyed Gypsy,
Will Ye Go Lassie Go,
Tis Pretty to be In Ballinderry,
Cruise of the Calabar,
As I Roved Out,
The Wee Falorie Man,
Rockin the Cradle,
I'll Tell My Ma,
I Wish I Was A Maid Again,
The Gallant Forty-Twa,
Early Early All In The Spring,
When I Was Single,
The Irish Girl
Fan-A-Winnow,

And then children and a woman singing
Three Grey Geese (and other children's tunes),
See Saw (and other children's tunes),
and Sally Go Round the Moon (and other children's tunes).

Subject: RE: Loving Hannah
From: Martin Ryan
Date: 12-Nov-98 - 04:43 PM

Alice

I thought that track listing sounded familiar! I have a Tradition CD reissue ("I am the Wee Falorie Man") of an album by Davy Hammond, of Northern Ireland, originally issued in 1959. It's identical! Only himself and and guitar though.Could it be him?