Annoyance was the feeling an individual got when someone or something made them feel angry or impatient. Irritation was one of the properties of annoyance.
In 2265, Spock assessed that, in time, a mutated Gary Mitchell infused with psionic energy would become annoyed with other beings. The implication of this assessment being that Mitchell would kill the crew of the USS Enterprise. (TOS: "Where No Man Has Gone Before")
In 2266, when he was bluffing his way out of a precarious situation, James T. Kirk said to his counterpart on the Fesarius, "We grow annoyed at your foolishness." (TOS: "The Corbomite Maneuver")
Later that year, Spock told Kirk that Leonard McCoy said that the captain had barged into sickbay acting like a "wild man" and demanded Saurian brandy. Kirk laughed and told Spock that the good doctor had been putting him on again. Spock apologized for the intrusion and Kirk said he would tell McCoy that the Vulcan was properly annoyed. (TOS: "The Enemy Within")