Hurt can refer to suffering or pain — physically or emotionally. The word is both a noun and a verb. The verb "to hurt" can mean either "to cause pain" (e.g. "You hurt my feelings") or "to experience pain" (e.g. "My bottom hurts").
Hurt and spanking
Corporal punishment is about the measured infliction of pain for the purpose of punishment. So a common complaint from spankees is "that hurt!" either as attempt to influence the spanker or simply as alternative to cursing is coping with the pain. A stereotypical response is often alone the lines of "It's a spanking it's meant to hurt".
Corporal punishment requires that a distinction be made between causing hurt and causing harm. where hurting is transitory and harm is lasting.