The questions are pretty much in the title.
Did Rasputin actually help the Tsar’s son’s disease?
If so, how did he do it?
The term is actually Tsarevich.
Nobody’s quite sure what Rasputin was able to do, but one opinion (as given in *Nicholas and Alexandra * by Robert Massie) is that he was able to use his forceful personality to create a sense of calm about the Tsarevich. This countered the “panicky” atmosphere that was produced by the continual examination of the doctors and helped the young boy to relax, thus aiding in the slowing of the blood flow.
Alexandra also fell very heavily under Rasputin’s influence. As a sick child, if your mother believed you were being healed, the power of suggestion would probably go a long way.
A wizard did it.
Actually, it’s Цесаревичъ (Tsesarevich).