You do have to consider the component of the pushing force on the lawnmower. If the normal force simply cancelled out the force of gravity, the net force in the vertical direction would be the downward component of the pushing force. This would mean the lawnmower accelerates into the ground, but this is clearly not the case.
What do we know in this situation? We know the gravitational force, and we can work out downward component of the pushing force. We need to relate the normal force to the things we already know. Using Newton's Second Law, we know that the net vertical force is 0, as there is no acceleration in this direction. The normal force is in the opposite direction to the weight force and the pushing force, so
This gives