to have or produce an itch

informal to have a restless desire

Perhaps she knew … that the chains she forged only aroused his instinct for destruction, as the plate-glass window makes your fingers itch for half a brick — Somerset Maugham

to cause (an area of the body) to itch

an irritating sensation in the skin that makes one want to scratch

informal a restless desire

A love of the theatre is so general, an itch for acting so strong among young people — Jane Austen