Combine potatoes, broth, carrots, and garlic in a large pot and bring to a boil, about 5 minutes. Stir in salt, garlic powder, and black pepper. Reduce heat, cover, and simmer until potatoes and carrots are tender, 8 to 10 minutes. Add corn and ham. Reduce heat to lowest setting.

While soup simmers, melt butter in a saucepan and whisk in cornstarch. Add milk and whisk to combine. Cook and stir until thickened and bubbly, 3 to 5 minutes. Add Cheddar cheese and stir until melted, about 2 minutes. Pour mixture into the pot with the soup. Cook over medium heat, stirring occasionally, until heated through, about 2 minutes more, or until ready to serve.