SciFi story where a man can miraculously alter things in real life by switching things between parallel dimensions

"What Rough Beast?", a novelette by Damon Knight, first published in The Magazine of Fantasy and Science Fiction, February 1959, available at the Internet Archive. You may have read it in one of these compilations.

Here's the part where he fixes the girl's burnt skin; a slow process, not a flick of the hand:

She jumped when hand touched her, but then sat still. I felt under my fingertips cold skin, touch like lizard. Inside me was big hurt jumping, I could not hold in very long. I rubbed her very easy, very slow with my fingers, looking and feeling where was inside the wrong kind of skin. Was not easy to do. But if I did not do it this way, then I knew I would do it without wanting, all at once, and it would be worse.

To make well all at once is no good. Each cell must fit with next cell. With my fingertips I felt where down inside the bottom part of bad skin was, and I made it turn, and change to good skin, one little bit at a time.

She sat still and let me do it. After while she said, "It was a fire, two years ago. Pop left a blowtorch lit, and I moved it, and there was a can of plastic stuff with the top off. And it went up—"

I said, "Not to talk. Not necessary. Wait. Wait." And always I rubbed softly the bad skin.

But she could not bear to have me rub without talking, and she said, "We couldn't collect anything. It said right on the can, keep away from flame. It was our fault. I was in the hospital twice. They fixed it, but it just grew back the same way. It's what they call keloid tissue."

I said, "Yes, yes, my dear, I know."

Now was one layer on the bottom, soft skin instead of hard; and she moved a little in the chair, and said small voice, "It feels better."

Under my fingertips the skin was still hard, but now more soft than before. When I pushed it, was not like lizard any more, but like glove.

[. . . .]

Under my fingers was a little place of good, soft skin, smooth like cream. While I moved my fingers, slowly this place got bigger. She looked down, and she forgot to breathe.

Here he tells us how he works his miracles by reaching into parallel worlds:

Since I was small boy in Novo Russie—what they call here Canada, but it is all different— always I could see where beside this world is many other worlds, so many you could not count. To me is hard thing to understand that other people only see what is here.

But then I learned also to reach, not with hands but with mind. And where this world touches other world, I learned to turn so that little piece of it would be different. At first I did this without knowing, when I was very sick, and frightened that I would die. Without knowing it I reached, and turned, and suddenly, I was not sick. Doctor was not believing, and my mother prayed a long time, because she thought God saved my life by a miracle.

Then I learned I could do it. When I learned badly in school, or if something else I would not like would happen, I could reach and turn, and change it. Little by a little, I was changing pieces of world.