Mightn't /ˈmaɪtnt/ is the contraction of might not. It is uncommon, especially in American English, where might not is the normal choice.
The apostrophe marks no squeezed to n't. The cited American dictionary transcribes the contraction as /ˈmaɪt nt/, where the /t/ and /n/ sit so close together that the whole thing can sound like one cluster. The Cambridge entry, on the other hand, shows /ˈmaɪ.t ə nt/, with a schwa between the /t/ and the /n/, so you may well hear a reduced vowel in there. Both renderings are fine: say it as a tight cluster or with a quick /ə/ tucked in between the consonants, and either matches a dictionary transcription.
We mightn't finish today, so let's plan for Friday. Writing might not apart is always safe.
Quick check: which contraction fits this gap: I ___ be able to come tonight. Good result: mightn't, the contraction of might not. Wrong result: mightn'te: the apostrophe already replaces the letters of not, so nothing else is added.
I mightn't be able to come tonight, sorry. A plain statement where might not would be equally fine.
You mightn't have heard, but the shop's closed Mondays. A soft aside where the contracted form keeps the tone light.
He mightn't agree, might he? Question tags built on might exist but sources note they are not very common. Here the tag might he? mirrors the verb in the statement, so the two forms sit naturally side by side.
Spelling and sound notes
A spelling trap: writers sometimes add an extra letter as mightn'te or drop the apostrophe as mightnt. Pronunciations differ by variety, and the vowel between the main syllable and the final /nt/ can be hard to hear. The American transcription /ˈmaɪt nt/ puts the /t/ and /n/ side by side, so the whole thing can sound like one cluster, while the Cambridge transcription /ˈmaɪ.t ə nt/ keeps a reduced schwa vowel before the final /nt/. Either way, the apostrophe marks the omitted letters of not, which reduces to n't, not a missing sound.
Restriction: this contraction is rare in American English, where might not is the normal choice. If you are unsure which variety your reader expects, writing the two words apart is always safe.