Melt the butter in a non-stick Dutch oven over medium-high heat, then add the leeks, carrots and celery. Cook, stirring frequently, until the leeks are softened, about 5 minutes. Add the potatoes, broth, water, thyme, salt and pepper; bring to a boil. Reduce the heat and simmer, covered, until the potatoes are soft, about 20 minutes.
Add corn and simmer, covered, about 2 minutes. Mash lightly with potato masher to break up the potatoes.