Spread half of the mayonnaise on one of the tortillas and place in the hot skillet mayonnaise-side down.
Top with half of the Monterey Jack. Add the corn, Cojita, avocado, jalapenos and cilantro. Sprinkle over the ancho chile powder and remaining Monterey Jack.
Spread the remaining mayonnaise on the other tortilla, then place on top mayonnaise-side up, and press. Cook until the bottom tortilla is golden and crisp, 2 to 3 minutes, then flip the quesadilla and cook until the second side is golden, an additional 2 to 3 minutes.
Transfer the quesadilla to a cutting board and slice. Serve with the salsas and tortilla chips.