Heat oil in a medium saucepan over medium heat. Saute onion, celery, and carrot until tender, 6 to 8 minutes.
Add potatoes and 4 cups water. Season with salt and pepper. Bring to a boil; reduce to a simmer. Cook until potatoes are tender, about 10 minutes. Mash half the contents of the pot until slightly thickened.
Add kale and beans and continue cooking until kale is tender, 6 to 8 minutes more.
Stir in vinegar and adjust seasoning if necessary. Drizzle with oil before serving.