Explain how John the Savage uses physical punishment as a way to try to cleanse and purify himself of both societies that he has known.

In chapter 18 of Brave New World, John the Savage mutilates his own body as a means of purifying himself. In his isolated lighthouse, for example, John whips his "rebellious flesh" and endures all kinds of physical pain and suffering.
For John, purification is necessary because of his experiences in both societies. In England, he fell in love with Lenina without really getting to know her. Then, on discovering that she had slept with many other men, he called her a "whore" and withdrew from her life. John, therefore, could not accept the sexual attitudes of this new society.
Similarly, in the Reservation, John bonded with and loved his mother, Linda. Arguably, he feels considerable guilt because of her soma addiction and death in the new world. John, perhaps, wishes that he could have done more to save his mother from such a fate.
John, therefore, mutilates himself to escape the disgust and betrayal that he feels about Lenina and the guilt he experiences after his mother's death.