Always difficult to speculate, but it could well be that a bit of tension crept into your feet and you inadvertently landed with a little bit of toe brake applied on one of the wheels. That might explain why you veered to one side. I like to make sure that the tips of my feet are on the bottom part of the pedals during the flare and landing, if that makes sense. Then when I want to brake I physically move my feet up the pedals to operate the toe brakes.
You also don't mention what the wind was doing, which could have been the factor that caused the aircraft to veer.
But once the aircraft had lifted itself off again, you did the right thing in applying full power and converting it into a take-off... it's never a good idea to push the aircraft back down onto the ground if it's trying to get airborne!
(Disclaimer: I'm only a lowly PPL myself, not an instructor. Probably best to chat about it with your instructor.)