It sounds like you are learning the reconstructed pronunciation system for Latin. This pronunciation system represents what we think ancient Latin speakers sounded like. Latin speakers used Latin sounds, not English sounds, so there are some situations where it is difficult to discuss Latin pronunciation by reference to English. English has weird vowel sounds compared to most other European languages. Latin spelling is mostly straightforward, but in a few areas Latin has multiple sounds that correspond to a single spelling. The reason for this is either because the sounds are related in Latin, or because Latin speakers naturally used different alternatives in different contexts.
The vowel in the last syllable of "lyricus" is a Latin short "u" sound.
This is pronounced the same as in the first syllable of a word like "puto" or "sunt". There is no exact equivalent in most accents of English; it's certainly not the sound most English accents use in words like "nut", "gust", or "fun", but it also isn't exactly the sound used in "boost", "moon", or the one in "foot", "put". In the International Phonetic Alphabet, the Latin short u sound is best represented by either [ʊ] or [u]. The sound [ʊ] is like the German short u sound; the sound [u] is like the Italian or Spanish "u" sound, or the French “ou” sound. (There is some disagreement about whether the Latin short u sound differed in Classical times from the long u sound in the quality of the vowel, or only in duration.) If you’re comparing to English, the sound of “oo” before “l”, as in “cool” or “fool”, is a bit closer to [u] in most accents than the sound of “oo” before other consonant sounds.
In the case of teneo, there is an argument, presented in W. Sidney Allen's Vox Latina, that any vowel before another vowel in Latin had the same quality as a long vowel. For "e", this means in the system Allen describes that the vowel had the quality [e] rather than [ɛ], despite being short. (The vowel [e] is usually approximated by English speakers as "ay".) However, this isn't necessarily an absolute phenomenon; it's possible some variability between both qualities existed, as Romance languages show some outcomes suggesting the quality [ɛ] was also possible in some cases (for example, French dieu, from Latin deum, has a diphthong ie, which generally goes back to [ɛ] rather than [e]).
Regardless of whether the quality of e in this position was [e], [ɛ], or either indifferently, it was pronounced with a short duration, and so counts rhythmically as a short vowel. (This is significant in poetry.) This is the reason why the vowel is not marked long.