Heat 3 tablespoons oil in a large saucepan over medium heat. Add onion and garlic and cook for 3 minutes. Add potatoes and cook, stirring constantly, for 3 minutes. Pour in water and bring to a boil; boil gently until potatoes are very tender, about 20 minutes.
Meanwhile, cook sausage in a large skillet over medium-low heat until it has released most of its fat, about 10 minutes. Drain.
Use an immersion blender to purée potato mixture. Stir in sausage and season with salt and pepper. Increase the heat to medium, cover, and simmer for 5 minutes.
Stir in kale and simmer until tender and bright green, about 5 minutes. Stir in remaining 1 tablespoon olive oil and serve immediately.