This is what is called "silent e" (there are lots of teaching materials that use this term). It's most often found in words with specific spelling patterns: you can see an overview in this answer: Silent "e" at the end of words. (That question wasn't actually the same as yours.)
The main rule about "silent e" that is relevant to "‑t" vs. "‑te" is based on the pronunciation of vowel letters earlier in the word. The spelling "‑te" is used immediately after a single vowel letter that is pronounced "long" (one that "says its name"). 'participate', 'Kate', 'electrolyte' and 'phosphate' all have "long" vowels before the final /t/ sound.
Words that don't have "long" vowel sounds in the last syllable are rarely spelled with "‑te", although there are certainly a few exceptions. Probably the biggest class of these consists of words ending in "‑ate" that are now usually pronounced with the schwa sound, such as pirate; there are also some words spelled with "‑ite" where the "i" is unstressed and not pronounced long, such as definite.
After a stressed "short vowel", as in cat, chat, let, it would be very unusual to find "‑te".
Likewise, it would be very unusual for an English word to be spelled with "‑xte" or "‑cte" rather than "‑xt" or "‑ct".