What is the verb used to mean the action of blowing air with your mouth because you are bored,annoyed etc? I couldn't find it on the internet. I need you guys' help. We call that sound made, "Üfff...," in my language. It sounds like "ffff." I don't think it is exactly a "sigh."
You could say they 'let out a huff'.
A 'huff' is a sharp exhalation of air. It can also be a verb 'to huff'.
It can be used in contexts other than expressions of annoyance, for example, if someone is out of breath due to exercise ("we were huffing and puffing by the time we'd climbed to the top of the hill").
When someone 'huffs' in connection with speech it indicates indignance, for example:
"Certainly not!" he huffed.
Even without associated speech, in a context where it is a reaction to something said or done, saying "he huffed" or "he let out a huff" would aptly describe an expression of indignance.
For contexts where it is less about annoyance and more boredom, you might consider saying they gave (or let out) "a heavy sigh"?