Learn English – What’s the word / way to describe the sound we make when we get irritated

Are you thinking of tsking? I realize that the action isn't the same, but I'm not sure there is a separate word for the specific mouth noise you're describing.

interjection, noun
(a sound) used to express disapproval, genuine or mock sympathy, etc.: a click, or sucking sound, made by touching the tongue to the hard palate and rapidly withdrawing it

Perhaps raspberries, as in to blow raspberries:

2 [short for raspberry tart, rhyming slang for fart] : a sound of contempt made by protruding the tongue between the lips and expelling air forcibly to produce a vibration; broadly : an expression of disapproval or contempt

There are many kinds of whistling. This site enumerates all these:

• Pucker Whistling: The pucker is the most common form of whistling. In this form of whistling, air is either expelled or sucked through pursed lips to produce turbulence and in turn, the whistle.
• Finger Whistling: It is also known as wolf whistling and the technique involves inserting fingers in the mouth to shape its opening. The whistle produced by this method is forceful and the sound is loud.
• Palatal or Roof Whistling: In this type of whistling, the air that is pushed between the roof of the mouth and tongue produces the turbulence.
• Hand Whistling: To whistle in this manner, the hands should be cupped together and the air needs to be blown out of the mouth into the resonant chamber formed by hands.
• Bottom-lip Whistling: The bottom-lip whistling is performed by sucking the air inside the mouth. The turbulence in this type of whistling is created by pinching the bottom lip at the center. The sound produced by this method is piercing and one should practice regularly in order to master this loud whistling technique.
• Teeth Whistling: It is the most difficult form of whistling and requires the whistler to draw air through the central incisors.
• Throat Whistling: The mouth is kept closed for whistling in this manner. In this method of whistling, the air is forced between the tongue and the back of the throat.

In my experience, what they are calling teeth whistling is more often used in the singular and so is called tooth whistling.