I'm putting this in the "New" category, because I'm not sure where else to put it.

"Tabhair dom do Lamh", aka "Da Mihi Manum" or "Give Me Your Hand", is an old classical Irish harp tune. It's called that because it was written as a plea for forgiveness by the composer/harper Ruairi Dall O'Cathain (aka O'Cain), to a lord he'd offended -- which worked very well. Anyway, it's a fine tune, and you can hear the pleading phrase "Da mihi manum" in the repetition of the phrases.

So my thought is that you could have either a short Kyrie for the OF by singing the first set of phrases as just:
Ky-y-ri-e
e-le-e-i-son
Chri-is-te
e-le-i-son
Ky-y-ri-e
e-le-e-i-son.

Alternately, you could do a very very melismatic Kyrie by spreading it out over the whole length of the first half of the basic tune. The second half is a bit more cheerful, so maybe you'd want to stop before then. Also, the usual way the tune is played is to speed up as you go, which you probably wouldn't want to do. You can do it in a very timeless and dignified sort of pleading way, and that's probably what would work for Mass.

Either sort of Kyrie would also work with the English, but you'd have to divvy up the phrasing a bit more carefully.

Probably this has already been done, but I just thought I'd put it out there in case it hasn't been. It's public domain, it's free, it's not hard to learn, so it's doable. I'll look for a score, but here are some examples of things that people have done with it:

Notre Dame Folk Choir throwing their resources at this fine tune. They do some variationy things.

A piano MIDI with a lot of extra doohickeys standing in for a harp bass line

The tune title is often misunderstood, and people actually use it as a wedding song these days. I certainly hope the loving couples aren't pleading for current forgiveness at that point.... The wedding song lyrics