Is it permissible to kill a venomous black cat that follows children, or is there another way to keep it away from the the house?

It is permissible to kill a cat if it causes actual or probable psychological or financial harm, especially if it is black, as jinns may take its form. Ibn Taymiyyah issued a fatwa (definition) permitting its killing if its harm cannot be repelled except by killing it. Al-Zayla'i also permitted killing harmful animals like "cats" if they eat pigeons and chickens, to remove their harm, and this should be done by slaughtering them, not by beating them. Excellence must be observed in killing. If it is possible to remove it without killing it, then it should not be killed. One must be certain that harm has occurred or that it is highly probable, and that killing is the only means of getting rid of it. The child's attachment to the cat may be due to the child's kindness towards it.