Either way, NASA is already exploiting it. I guess, next they’ll find a way to glitch through the very fabric of the universe to teleport to a distant galaxy without moving at all or even using any energy.
In short, even though photons have no mass, they still have momentum proportional to their energy, given by the formula p=E/c. Because photons have no mass, all of the momentum of a photon actually comes from its energy and frequency as described by the Planck-Einstein relation E=hf.
But how do you apply this with Lorentz' transformation (i.e. relativistic factors)? You cannot approach the speed of light without considering relativism. It is known that p = gamma * m * v where p is momentum, gamma is the gamma factor given by sqrt(1/(1 - (v^2/c^2))), m is mass and v is velocity. If you study the gamma factor, you'll realize that it approaches infinite as v approaches c, the speed of light. Since we are actually dealing with light here, where v = c we are breaking the equation. Momentum cannot be defined for any mass which moves at the speed of light. It's asymptotic at that speed.
Also note that the same goes for E = mc^2. At relativistic speeds, also this equation needs to consider the gamma factor. So those classical equations break down for light.
The answer is that photons don't have mass, but they have energy. There is a good explanation a bit further up in this thread on how this is possible.
They can also be created or absorbed into something else. The mass of whatever absorbs them increases, and the mass of whatever is emitting them decreases when they do that.
The mass of everything is changing all the time. The thing that is constant is the rest mass.
The object doesn't absorb their mass, but rather their energy (which admittedly can be equated to a mass via a factor of c^2, but that's not actually what's happening). The change in momentum that results from a photon hitting you isn't caused by a change in m, it comes from a change in v. If mass were the quantity being transferred, solar sails wouldn't work to move anything; they would just sit there and get more massive as photons hit them.