to the best of my knowledge, Fáinne Geal an Lae is the song from which Paddy Kavanagh got the tune for his poem. I don't know what tunes the other lyrics given above are set to. (there are both other translations of Fáinne Geal an Lae above and different songs with the line "dawning of the day" in them)

THE DAWNING OF THE DAY
"Fáinne Geal an Lae", anonymous, 18th century
Poetic translation by Edward Walsh

At early dawn I once had been
Where Lene's blue waters flow,
When summer bid the groves be green,
The lamp of light to glow.
As on the bower, and town, and tower,
And widespread fields I stray.
I met a maid inthe greenwood shade
At the dawning of the day.

Her feet and beauteous head were bare,
No mantle fair she wore;
But down her waist fell golden hair,
That swept the tall grass o'er.
With milking-pail she sought the vale,
And bright her charms' display;
Outshining far the morning star
At the dawning of the day.

Beside me sat that maid divine
Where grassy banks outspread.
'Oh, let me call thee ever mine,
Dear maid,' I sportive said.
'False man, for shame, why bring me blame?'
She cried and burst away -
The sun's first light pursued her flight
At the dawning of the day.

Notes about the translation from Kathleen Hoagland,ed., "1000 Years of Irish Poetry". Old Greenwich, Conn.: Devin-Adair, copyright 1927, renewed 1975 - out of print :
"Walsh, Edward - was born in Derry in 1805; died in Cork, August 6, 1850. Walsh was well educated and became a private tutor and then a teacher in Cork. Moving to a smaller town, he began to write for the magazines and later went to Dublin. From Dublin, he was appointed schoolmaster to the convict station at Spike Island in Cork Harbor, now Cove. From there he moved to Cork Workhouse, where he taught until his death. He was a contributor to the 'Nation'. His work includes two volumes of translations from the Irish - the original text and then the translation. He was considered, next to Crofton Croker, as the authority on legendary and fairy-lore of the country."

I would very much like to see those "two volumes of translations from the Irish - the original text and then the translation." Hoagland's book contains a large number of poems in English that derive from older poems in Irish, but only the English-language versions are presented. But Fáinne Geal an Lae is easily obtainable from other printed sources. Here are two verses from "Amhránleabhar Ógra Éireann", Dublin: Folens & Co., 1971 (8th edition):

Fáinne Geal an Lae

Maidin mhoch go ghabhas amach
Ar bhruach Loch Léine
; An samhradh 'teacht 's an chraobh len' ais
'Gus lonradh te ón ngréin.
Ar thaisteal domí bhailte poirt
Is bánta míne réidhe,
Cé a gheobhainn lem ais ach an chúileann dheas
Le fáinne geal an lae.

Ní raibh bróg ná stoca, caidhp ná clóc'
Ar mo stóirín óg ón spéir,
Ach folt fionn órga síos go troigh
Ag fás go barr an fhéir. Bhí calán crúite aice ina glaic,
'S ar dhrúcht ba dheas a scéimh;
A thug barrghean óVéineas deas
Le fáinne geal an lae.

Verse 1 - I went out early morning by the backs of Loch Lene/summer was coming, trees budding, warm rays of the sun/travelling through harbour towns and meadows, /who should I meet but a pretty fair maiden/at the bright ring [dawn]of day.
Verse 2 - My young treasure from heaven wore no stockings, cape of cloak/ But her long golden hair swept the grass/ She clutched a milking-pail /and she looked beautiful in [morning] dew/that brought love from Venus[?]/With the bright dawn of the day.