C172 flight Instructor reported their student while landing became airborne and drifting off the runway centerline landing again with full brake resulting with the aircraft leaving the runway hitting a few lights then entering a taxiway.

Narrative

Student came in for a normal landing had a stable approach. Student touched down and had 3 wheels on the ground. (crosswind wheel first; followed by the other main; and then the nose wheel.) I could visually see the right rudder in. Around 3 seconds later; I heard an increase in power but was not full power; At this point; I was wondering what was going on. student had put the ailerons more into the wind since they were all the way on the ground (and on centerline); but because they added power they were producing enough lift again to rotate. student pilot was about a foot or so off the ground and started to drift to the left; and he began to push down to get back on the runway; and was applying full brake when he touched down; and began going off the side of the runway hitting a few lights. I told the student to put power idle (over the radio); and it took a little time for him to react. This allowed him to start dissipating energy to keep him on the ground. He began going off of the runway environment and off toward the taxiway went thru the grass back on to the taxiway.Student pilot should have gone around.