Is breathing through your mouth in cardio workouts the correct way, or does more pollen hit the lungs?

No right or wrong. While you may get a little more pollen in your lungs with mouth breathing than nose breathing, you can't sustain high ventilation through your nose for more vigorous cardio, and the difference in the amount of pollen probably makes little difference to your lungs. Getting exercise is more important than how you breath. Let your body tell you how to breath, and don't think about it.