In the Roman Rite in the USA, we'd normally use Italianate Church Latin pronunciation, not, say, English or German Latin. Comparison:

https://en.wikipedia.org/wiki/Latin_regional_pronunciation

I would say most people in the USA pronounce it "clehmens" but soft vowels are almost non existent every where else- however when Americans pronounce a hard vowel its often much more exaggerated. Thus, technically speaking "claymens" but its a very slight emphasis. You should shoot for somewhere in between if its Americans we are dealing with.

I have been told many times, the main problem with American pronunciation is that we are lazy with the shape of our lips.

The functional reason many American church choir directors try to avoid unnecessary diphthongs like "clay" is that they have a tendency to go flat in the mouth of a typical volunteer American church chorister.

As a Brit, I don't know how you pronounce 'eh'. When it stands alone, as it does in English colloquial meaning "I didn't catch what you said" it rhymes with 'pay'. As indeed does Salve.
http://hull-awe.org.uk/index.php/Pronunciation_of_Ecclesiastical_or_Church_Latin

But there must be exceptions to always pronouncing "e" as "eh". What about "Salve" or "Christe"?

E is never a diphthong in Church Latin.

So, I contacted one of our Sisters who actually has her doctorate in Latin. And she sent me this:

Vowels
a is pronounced as in father: sanctam (sahngk-tahm).
e is pronounced as in egg: ante (ahn-teh).
i is pronounced as in machine: filii (fee-lee-ee).
y is pronounced the same as i: Kyrie (kee-ree-eh).
o is pronounced as in tone: omnia (ohm-nee-ah).
u is pronounced as in ruler: unum (oo-noom).
Note: When two vowels appear together, each is pronounced: mei (meh-ee). (This rule does not apply in the cases of certain diphthongs—see below.)

Diphthongs
ae and oe are pronounced like e: saeculum (seh-koo-loom).
au and eu are treated as single syllables, but each vowel is pronounced distinctly.
In singing, the first vowel is sustained, as in other combinations of two vowels:
lauda (lah-oo-dah).

Consonants
The consonants b, d, f, k, l, m, n, p, s, t, and vare pronounced as in English.
c before e, i, y, ae, oe is pronounced ch: coelo (cheh-loh); in all other cases,
c is pronounced k: cantus (kahn-toos).
cc before e, i, y, ae, oe is pronounced tch: ecce (eht-cheh).
ch is pronounced k: cherubim (keh-roo-beem).
g before e, i, y, ae, oe is soft (as in gel): genitum (jeh-nee-toom); otherwise, g is hard (as in go): gaudeamus (gah-oo-deh-ah-moos).
gn is pronounced ny: agnus (ah-nyoos).
h is mute, except in special instances, when it is pronounced h as in English (however it is more like a mix between h and K close to the Spanish J: mihi (mee-hee) and nihil (nee-heel).
j is pronounced as y: Jesu (yeh-soo).
qu is pronounced as kw: qui (kwee).
r is lightly rolled with the tongue.
sc before e, i, y, ae, oe is pronounced sh: ascendit (ah-shehn-deet).
th is pronounced as if the h were absent, as in Thomas.
ti before a vowel and after any letter except s, t,or x is pronounced tsee: gratia (grah-tsee-a).
x is pronounced ks: ex (ehks).
xc before e, i, y, ae, oe is pronounced ksh: excelsis (ehk-shehl-sees).

while Italian choirs are taught that there are only 5 Latin vowels, in actual practice they employ the 7 of their vernacular

None of which is a diphthong. Ay as in clay is.

Do you mean clemens with the first e pronounced like clam? (very different from clay)

I think Henry is right about /klɛmɛns/ and /Kriste/, with /klemɛns/ being the German variant.

From the Oxford Classical Dictionary entry "Pronunciation, Latin":
Whilst the short ă and the long ā were of similar quality, the other long vowels were distinctly higher than their short counterparts. In fact, in the Romance languages, the mid-long ē and ō developed in the same way as the high short ĭ and ŭ. The approximate values of these vowels in terms of modern languages are as follows: ă/ā as first/second vowels of Italian amare; ĭ/ī as in Eng. bit/beat; ŭ/ū as in Eng. put/fool; ĕ as in Eng. pet; ē as in Fr. gai or Ger. Beet; ŏ as in Eng. pot; ō as in Fr. beau or Ger. Boot.

Note how the authors of the article (W.S. Allen and J.G.F. Powell) were hard-pressed to give an English word that has the sound ē. As a U.S. English speaker, it is virtually impossible for me to pronounce the sound without adding a -y to it and pronouncing it as a diphthong. In the Latin word clemens (nom/voc sing), both vowels are long. Be careful, as in other forms of the word, the e after the m is short (clēmĕntem).

The aspirated consonants k (also hard c and qu), p, and t are pronounced with a strong puff of air in good English or German diction. In church Latin, we should use a dental t, just as in Spanish, Italian, or French. It's not a d. More like a d? Yes, somewhat, but t is not a voiced consonant. Qui tollis peccata doesn't become gui dollis beggada, but the hard aspirated consonant sounds of English or German are out of place. I'm afraid they're ubiquitous among British choirs singing in otherwise Italianate Latin, and the situation isn't much better on this side of the pond. Terra is a good example because the word descended unchanged from Latin to Italian.

The rolled r is another problem for English speakers, as are avoiding ay diphthongs instead of a pure [ɛ] vowel (coupled with hypercorrection of the real diphthong where it occurs, such as ejus becoming eus and eleison become elesson), and au coming out more like [ɔ] than a diphthong. Intervocalic s is another can of worms. The Italians don't really say Jessu or vissibilium, but the voiced z sound is sometimes considered an "informal" pronunciation. German speakers have no difficulties with [ɛ] or [aʊ̯], but in my limited experienced, they have a tendency to forget that h is silent when attempting Italianate Latin, and semper and other -er words come out rhyming more with uh than air.

I never encountered ecclesiastical Latin in the Latin classroom!

I did encounter the dental T (then again, I preceded my Latin class in college with 9 years of Spanish starting at age 9 - my public school district required foreign language instruction to begin early and be sustained - by the time I was in high school, I had occasional dreams in Spanish).

Also, I grew up in a family with parents from Connecticut who at the time still had that fairly neutral (by US accent standards) Western New England accent. I learned Latin in Charlottesville, where the Piedmont American English accent ran riot (very easy to pick up inadvertently, and I can readily lapse into it if I am talking with college friends even at a 45 year remove - the danger of learning multiple languages young!) along with Tidewater and other sub-regional accents, so I suspect the professor picked his battles accordingly.

This reminds me that people use diphthong to mean digraph when it refers to vowels that were diphthongs but became /e/ (or some other singular vowel sound) in later forms of Latin and in ecclesiastical Latin.

/klɛmɛns/ is what I ask for sometimes versus /klemɛns/ because I’d get /ai/. Laus in Ecclesia gives examples like maid when it is trying to get you to sing the close (really close-mid) vowel in a word like Deus. And then it gives pen but that didn’t account for pin-pen merger. Red works though.

We struggle with the hyper correction of the semivowel to /u/.

I second the comment about the /t/. I don’t do anything but lighten the /r/ because we have a native Portuguese speaker, a Miami English speaker with some Spanish influence on his English, myself who unfortunately learned the /r/ of French but never could manage it in Italian or Spanish as an alveolar tap or trill, and the rest of the bunch can’t do any of them.

Eleison is only shortened when the polyphony calls for it. We’re good about that one.

Now I have to watch my /b/ lest it become /p/ like at the beginning of the Requiem tract. But I do think that having had experience teaching language helps; I do think of lighter French vowels as my reference point: /ɛ/, /œ/, /ø/, and /ə/ (and, with the middle two you also have /ɔ/ for that matter) sometimes hard to distinguish. And luckily most of those are close to English sounds.

I’m not going for a super-rounded /u/ for example, like a French kid who says hello to an owl (Coucou, hibou !) but it’s pulled back a bit, if that makes sense…