Will pushing on a car always change the car's mechanical energy?

I assume you mean that the car's motor is not running ... the car is just
sitting there.

If that's so, then the car's mechanical energy is just like the mechanical
energy of any other object.  It has potential energy if it's in a high place
from which it can roll or fall, and it has kinetic energy if it's moving.

-- If you make the car move by pushing it, then you gave it kinetic energy
that it didn't have while it was just sitting there.

-- If it's already moving slowly, and you're able to make it move faster by
pushing, then you increased its kinetic energy.

-- If you're able to push it up a hill, no matter how small the hill is but just
to any higher place, then you gave it more gravitational potential energy
than it had before you came along.

In all of these cases, if you exert a force and keep exerting it through some
distance while the car moves, then you have done "work", which is just
another name for mechanical energy, and your work adds to the mechanical
energy of the car.

But if you didn't move the car, then no matter how hard you pushed, no work
was done, and the car's mechanical energy didn't change.