All right, this one’s not mine, I swear!
I got this out of "Treoir", Comhaltas Ceoltoiri Eirann’s magazine.
The caption relative to this tune indicated that it had been transcribed by Dennis Murphy in 1949. That’s pretty much all I know of it.
You’ll forgive me for not having resisted the temptation to include it here :-) .
"Fanning’s Jig" is in Ceol Rince na h’Eireann volume 5. Breathnach got it from a manuscript written by the great Sliabh Luachra fiddler Pádraig O’Keeffe in his fiddle tablature notation. O’Keeffe taught Denis Murphy and his sister Julia Clifford.
Denis Murphy recorded "Fanning’s Jig", as did Jackie Daly. According to Breathnach, this tune is related to "Clancy’s Jig": "…recorded by the accordion player P. J. Conlon - from the vicinity of Milltown, in County Galway - in New York in September 1928."
Other names for "Clancy’s" are "Katy’s Rambles" and "The Ladies’ Triumph", in American publications of the 19th century.
(Thanks to Paul de Grae for translating Brendan Breathnach’s annotations. See http://www.nigelgatherer.com/books/CRE/ )
Lovely jig, find it here about 1m 25s in.
FWIW, Josephine Keegan includes this tune on peg 126 of her tune collection "A Drop in the Ocean" under the title Fergal French’s Jig no. 2, also including another tune as Fergal French’s Jig no. 1 .
Quick followup — Alan Ng, at https://www.irishtune.info/tune/3135/ reports that Fergal French disclaims both these attributions.