Touch-and-go landings were being practiced, and the winds were variable. After power was added and the airplane was accelerating, the instructor noticed the student was not compensating for torque or crosswind. The instructor prompted the student to add right rudder, but the student reacted quickly with left rudder and brake. The instructor pulled power but did not want to make aggressive control inputs due to the airplane's speed. The airplane veered left, departed the runway, hit a runway light, and the nose gear collapsed.