Is it permissible to kill a dog that the mother keeps at home, which is not a guard or hunting dog, and is white in color, knowing that the mother does not pray because of it, and can this problem be solved?

It is not permissible to kill a dog unless it is harmful or vicious. If your mother has no valid reason to keep it, advise her gently to stop keeping it and explain the ruling on abandoning prayer and its seriousness. Continue advising her and do not despair. If you can take the dog and move it to another place without your mother knowing and without harming her, then do so to forbid evil. Be sure to supplicate much for her guidance and righteousness.