In 12-inch nonstick skillet heat 2 tablespoons oil over medium-high heat until shimmering. Stir in onion, corn, and poblano and cook until softened and edges of onion begin to brown, about 6-7 minutes.
Stir in beans, water, lime juice, and cumin and cook until water evaporates, about 1 minute.
Remove mixture from heat, and coarsely mash mixture with potato masher, and then season with salt and pepper to taste.
Spread ¼ of corn mixture over half of each tortilla, leaving ½-inch border around edges. Sprinkle ¼ of cheese mixture over corn mixture, and fold other half of tortilla over the half with filling. Press firmly to seal.
Wipe skillet out with paper towel, and heat 1 ½ teaspoons oil in empty skillet over medium-high heat until shimmering. Cook 2 quesadillas at a time until crispy and golden brown in spots, about 1-2 minutes per side.
Move cooked quesadillas to cutting board, and repeat with reaming 1 ½ teaspoons oil and last 2 quesadillas. Cut quesadillas into wedges and serve with sour cream, salsa, guacamole, and sweet corn cake.