I read this in at least two different short story collections (American, in English). It was probably originally published in the late 80's or very early 90's, when I first read it.
The story concerns a young man with psychic powers who can influence other people's minds. He's either injured or ill, in some kind of distress in the beginning, and is rescued by a middle-aged woman who runs a home for unusual people. She is an ordinary person with no powers, herself. The other residents include a sentient radio, a very young girl with firestarter powers, an elderly Spanish-speaking gentleman whose powers somehow keep the world from coming to pieces, and a shapeshifter who lives in the attic. This last being, Toby, is compelled to take the shape of whoever the person it is with most desires. The woman who runs the home has been working with it to try and help Toby develop an identity of it's own, but it is still very vulnerable and mostly stays shut up in the attic.
At first the psychic young man tries to fit in and help out around the house, but he soon starts causing problems. They are always short on money, but the woman in charge is disturbed when he "pays" for groceries by wiping a cashier's memory of them. She finds it difficult to get him to understand why this is morally objectionable. He also develops a selfish interest in Toby, and begins subtly forcing the shapeshifter to take on the form of his ex-girlfriend, with whom he is still obsessed. When the woman in charge objects, he gets angry and uses his powers on her.
I forget how the story ends; I think the woman is forced to kill the psychic. I do remember it is a tragedy, and the detail of the sentient radio being broken and, when repaired, is just an ordinary radio.