Final answer:
The nurse should document the burning sensation in the client's amputated limb as phantom pain, which is the sensation of pain from a missing body part.
Step-by-step explanation:
The nurse should document the client's sensation as phantom pain. Phantom pain is the perception of pain or discomfort that seems to come from the missing limb. It is different from referred pain, which is the perception of visceral sensation projected to a different region of the body, such as how a heart attack may cause pain in the left shoulder and arm. Phantom pain is a common experience for individuals who have had a limb amputated.