a light wave strikes the moon and reflects twards Earth. As the light travels to Earth does the wave carry energy and matter, just one or neither?

These fields have electric and magnetic energy, therefore the light wave carries energy.

it does not transport matter

Explanation:

Light is an electromagnetic wave formed by the oscillation of electric and magnetic fields.

These fields have electric and magnetic energy, therefore the light wave carries energy.

But the wave does not need matter for its movement, which is why it transports matter. If we use the criterion of special relativity the mass of a light wave that travels at speed must be zero, so with this criterion it does not transport matter either