Well you've now added additional torque to the torque already there from gravity. So will stay in the same direction, but increase in magnitude. Therefore dL/dt will increase, causing the wheel to momentarily rotate around more quickly.
If you tap in the horizontal direction, you add a component to the torque in the vertical direction. So is now no longer in the horizontal direction and the wheel will start moving up, or down, depending on the direction you tap it.