In a medium microwave-safe bowl, combine the chilies and water. Cover with vented plastic and microwave 2 min. Let stand 5 min.
To a blender, add the chilies and water along with the green onions. Blend until smooth.
In a large saucepot, heat the oil on medium. Add the garlic and cumin. Cook 2 min., stirring often. Add the broth, hominy, chicken, chili mixture, and sugar. Partially cover and heat to a boil on high. Reduce heat and simmer 10 min., stirring occasionally. Season with salt to taste.
Garnish with thinly sliced radishes and finely chopped jalapeños for fresh crunch.