Ahh...that makes more sense, like Hanukkah/Chanukkah. It's a voiceless velar fricative. Which explains why I would never have heard it--it's probably pretty difficult to tell an h from a ch in a choir. The k though is a different matter, as it's a velar stop and the airflow is totally constricted. I still can't imagine choirs sing a full stop there.
Speaking of which, I've noticed that some American choirs tend sing the K in Kyrie as a long, loud fricative: "KKKKHHHHHHyrie". That's characteristic of American English, where initial /k/ is aspirated: (if you hold your hand in front of your lips when you say "cat" and "scat" you can usually feel a stronger puff of air for the first one). In Italian, that isn't done; inital /k/ is nonaspirated.
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Since our choirmaster is big into the idea of not stopping the flow of air, that makes sense (and boy is it causing me trouble, because my transition from chest to head voice has always been lazy, with a click or glottal hesitation on a leap of more than 2-3 steps . . . old habits die hard, but we're working on it . . . arggggh!)
If you sing a Greek upsilon instead of an English "EE" in Kyrie, it moderates the initial K quite nicely . . .