I have a recording of this song by Brendan Moriarty on The CD "Bogman's Pipe" by The CELTS
Owl Records [911] CD 1998. The tune [with lyrics and tabs] [melody line only] is published in The Celtic Fake Book, "C" Edition, Hal Leonard Corporation, Milwaukee WI [ISBN 0-634-01727-6] [www.halleonard.com].

      County of Mayo - Tabs

(A)ON the (E7)deck of Pat- rick (A) Lynch - 's boat I sat in (D)woe- ful Plight
Through my(E7)sigh - ing all the (A) wear - y day and (E)weep-ing all the (E7)night
Were it (A)not that full of sor - row from my (E7)peo - ple forth I   (F) go,                                 
By the (G) bless - èd sun! 'tis (A) roy - al - ly I'd sing thy   (D)praise,May - (A) o.

X: 65
T:The County of Mayo
M: 4/4
L:1/8
Q:192
S: Celtic Fake Book "C" Edition Hal Leonard
K:A
z6EF|G2B,2C2D2|E3DC2B,C|DCB,CA,2A,2|
A,3z3CD|E2E2A2B2|c2c2dcBA|
G2FGE2E2|E3z3CD|E2E2A2B2|
c2c2dcBA|G2FGEFGE|A3Bc4|B2A2=G2B,2|
C2D2E3D|C2B,CDCB,C|A,2A,2A,6||

I think the melody sung by Brendan Moriarty is much smoother than the Fake Book version.

X:66
T:The County of Mayo
M:4/4
L:1/8
Q:192
S: CELTS The Bogman's Pipe adapted Warsaw Ed
K:C
z6EF|G2F2E2c2|B2BD2zDE|
F2D2C2B,2|C4z2EF|G2GA2zBc|
d2de2dcB|G2Bcd2zB|c4z2EF|
G2G2A2Bc|d2d2e4|dcB2G2Bc|
d2Bc4z|EFG2FE2z|c2B2D2DE|
F2D2C2B,2|C4||
                              Enjoy, Ed