"Would it be wrong to think that the KHA sound originally (in Proto-Turkic)"
Probably wrong because the []x sound at the beginning of those Tuvan words doesn't come from an original Proto-Turkic []x, but from a Proto-Turkic []k. A lot of the "Sayanic" Turkic languages softened a []k to []x. You can even find both []k and []x sounds "matched" in Tuvan alliterative lyrics and poetry (where the same sound is used at the beginning of each line).
I've just something proposed by Prof.Dr. Bahaettin Ögel.
According to him:
Ancient Turks differentiated the words KA and KAP.
"Ka", was a cup which contained liquid or solid food.
On the other hand, "Kap" was a cup which anything could be put inside. A food plate, a sack, a pillowslip and even according to Mahmud Of Kashgar, a woman's womb is a "kap" as well.
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Kacak and ka are only kitchen utensils.
In Kutatgu Bilig:
Neçe ka tegürgü kerek bu aþýg,
Kurug kodmasa bir içig hem taþýg
(How many cups of food should be given would be known... As a result nobody inside and outside will be left thirsty and hungry) (I guess it's close enough translation?)
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In Altun Yarug: There's something about a feast.
"öküþ telim kaplarta, tolguklarta" (Food and drinks were presented (to be presented?) in cups and tulums (cups made of fabric)
....
Turkish: kap (to catch and contain) English: catch
Turkish: kapa (to close and cover)
Turkish: kavra
English: cover
French: couvrir
Italian: coprir
Latin: coprire
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Capote -the thing that closes on the engine and covers it. (think of Turkish "kapat" : "close it"
Capture- The take someones freedom. (Kaptýr: "to unwillingly allow someone to capture something of yours" in Turkish)
Captive- someone captured (Kapýl: to be captured in Turkish)
Capacity- The capacity of a cup
Cabinet- The cup in which ministers are collected inside.
Capillery-Saç kadar ince kan damarý, kýlcal
Capital- Big letter. The first letter that contains the other small ones after it.
Capital-Baþkent, kafa kent
Capital- Sermaye, Capitalism, Capitalist
Cap-Denizde bir kara çýkýntýsý, kafadaki burun gibi, karaburun gibi