What's the word / way to describe the sound we make when we get irritated?

The action is called "sucking your teeth."

I don't believe there is a single-word for that sound specifically, it's just a "sucking sound."

The sucking sound of him sucking his teeth in irritation echoed in the empty room.

you mean the what was that, how did that happen, i meant to stay quiet kinda type? That can be done on purpose, in which case it might go as "smack ones lips", perhaps in immitation of sucking, but--as comics have it--involving licking (not so in German schmecken or schmatzen).

If it would be more often associated with the (tip of the) tongue, I think of German schnalzen, onomatopoetic e.g. tsk, tse. Wiktionary translates that as

click

under ety. 1 def. 2, giving tsk specificly as example (although African languages like xoo! are better known for their click-sounds).

In my experience similar noises can come from the uvular ridge, which then sounds like a bit of a grunt, but not oinky.

In general, these are implosive, or properly: ingressive. Which is rather rare as phoneme feature (often in Finnish for a specific yes expression, and similarly in many other languages). If there's no obstruent involved, it comes out a gasp, an intake of air, huh. There are more colorful idioms to describe these, I'm sure

When we press both our lips together and make a sucking sound, with our teeth closed.

That's called humph (or harrumph).

Humph (exclamation): a short, deep sound made with the lips closed, expressing anger or doubt, or pretended anger.

Example: Humph, I see you've got yourself some lunch and you haven't made any for the rest of us!

[Cambridge English dictionary]

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The sound made by tut-tut or tsk-tsk is a Dental click. Is that what you're looking for?

The tut-tut! (British spelling, "tutting") or tsk! tsk! (American spelling, "tsking") sound used to express disapproval or pity is a dental click. [Wikipedia]

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Grunt is the verb for making that sound.

Grunt (verb): (of a person) to make a short, low sound instead of speaking, usually because of anger or pain.

Examples: He hauled himself over the wall, grunting with the effort.

[+ speech] "Too tired," he grunted and sat down.

[Cambridge English dictionary]

Groan and its synonyms could also be used here.

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I think that's also called wincing or hissing (or chuff, cluck or squirm, but I'm not sure about these three).

Wince (verb): to tighten the muscles of the face briefly and suddenly in a show of pain, worry, or embarrassment.

[Cambridge English dictionary]

(I've also heard don't kiss your teeth at me in British English.)

There is too, a word for it in Sranan: "Tjoerie"/"Tyuri" (phonetic: "Chooree"). This is the language spoken in Suriname, the largest Dutch-speaking country in the Caribbean.

To my knowledge there isn't A word for it in English, which only has a description for the action: "kissing one's teeth".

In both environments, it's considered rude to do it. Pupils get berated if they do it in front of teachers, authority figures.

It's a bit neo-colonial, thus rude, to divide the Caribbean into English-speaking, French-speaking, Dutch-speaking and Latinx, therefor we should be careful whenever we do that: always check with the persons you speak with how they feel about that. But, IF we would do it in this case, then one could say that the Anglo and the Dutch have this custom/practice, and the French-Caribbeans call it 'tchip'. Afaik the Latinx don't do it, but my knowledge is not universal.

According to this article Brazilians have it too, but that's just the sound, sort of, with an entirely different meaning.

Tjoerie or Tyuri