Light waves exert their own gravitational pull and must be emitting gravitational waves, losing energy in the process. Does this mean that light becomes red-shifted as it travels even without the effects of universal expansion?
First we must clarify:
gravitational waves can be created by constantly changing gravitational fields, like two oscillating neutron stars.
photons do have gravitational effects, because they have energy
gravitational effects are because of stress-energy
photons energy comes from their frequency, E=h*f
a simple mass does not emit gravitational waves, like the earth, or at least not so big to have any effect.
a traveling photon does not emit gravitational waves.
gravitational waves are not the same as the effects of gravity (bending of spacetime), because gravitational waves are traveling gravitational fields, independently from the emitter
a gravitational wave is a traveling change in the bending of spacetime, so like a traveling gravitational field
a simple mass that has gravitational effects, like bending spacetime, does not emit gravitational waves (it needs to be rotating very fast around another mass, like two neutron stars rotating).
A simple mass like that will not lose energy by creating a gravitational field, and bending spacetime.
That is why a traveling photon will not lose energy by creating a gravitational field and bending spacetime locally where it is.
Photons get redshifted because the ones coming from far away galaxies that are receding, are in places where space expands, and the photons travel in expanding space that is why they get redshifted, getting bigger wavelength.
This does not directly answer your question about redshift, but does clarify some points: Do photons and cosmic rays radiate energy through gravitational waves? If not, why not?. A key point here is that the photon must be accelerated for it to emit a gravitational wave, a photon simply travelling through empty space from another star to earth would not be accelerated (due to no massive bodies around) and therefore would not emit any gravitational waves.
In essence, if the photon does emit a gravitational wave, due to passing near a massive object, then yes it would lose energy and therefore be redshifted. The amount would likely be immeasurable I would assume (though I cannot prove it).