What is it called when people who speak with a Connecticut accent cut off "t"s at the end of words or skip them in the middle of words?

Solution 1:

Glottal stop, according to the following article:

The glottal stop or glottal plosive is a type of consonantal sound used in many spoken languages, produced by obstructing airflow in the vocal tract or, more precisely, the glottis. (Wikipedia)

From The New York Times- Connecticut - Accent? What Accent?

"It's called a glottal stop," said Pat Gomola, a speech pathologist at the Speech and Language Institute in Middletown. "It's not a 't' sound. You say it in the back of your throat. It's the same thing when they say double-t words like cattle or bottle." In New Britain, such words come out as "CAH-uhl" or "BAH-uhl." Ms. Morgenstern attributed this sound to New Britain's large Polish-American population. "They don't enunciate their consonants as much," she said.

Solution 2:

I think there are a couple related but slightly distinct things that you're talking about.

At the end of the word Connecticut, you are probably talking about replacement of /t/ at the end of a word with a glottal stop [ʔ]. This also can affect /t/ before a consonant in the middle of a word, as in wetly.

In words like important, the relevant sequence is even more specific: you're talking about the pronunciation of /tən/, a /t/ followed by the unstressed vowel phoneme called "schwa" and then the nasal consonant /n/. In many accents of English, the phoneme sequence /ən/ can be pronounced as a syllabic nasal, transcribed [n̩], in certain contexts, including after /t/. The realization of the /t/ itself is a little variable: it can be a nasally released /t/ sound (transcribed [tⁿ]), or for some speakers it can be a glottal stop [ʔ].

As far as I know, the use of [ʔn̩] is fairly widespread, so it wouldn't stick out too much. A slightly different thing that I think you might be noticing is the use of oral release in this context. I said earlier that /tən/ is often pronounced with [n̩], but some speakers do in fact use something like [ʔən], with a glottal stop followed by an oral vowel followed by a nasal consonant.

Speakers who don't usually use oral release seem to find that the pronunciation of /tən/ with oral release sounds odd, but they describe it in different ways. I've seen some people say it sounds to them like the insertion of an /ɛ/-like vowel; another common perception seems to be loss of the /t/ sound. A recent Reddit thread on this subject: https://www.reddit.com/r/linguistics/comments/fq3vfz/millennialzoomer_glottalization/

A post there by problemwithurstudy links to the paper "Where are the mountains in Utah?", by David Eddington, which studies the topic of how /tən/ words are pronounced in Utah. Eddington found that what is popularly characterized as "dropping t's" in Utahn pronunciation is actually the use of a glottal stop with oral release. I don't know what the situation is in Connecticut but I wonder if it could be similar.

Decapitated Soul's comment "'Mountain' pronounced with a 'glottal stop' sounds awkward" reminded me that /ntən/ can behave differently from /tən/ for some speakers, but I don't know the details. Here's a 2016 Language Log post by Mark Liberman about the pronunciation of the name "Clinton": https://languagelog.ldc.upenn.edu/nll/?p=27112