Another thread on this site established that the photon has a gravitational field of its own. Then the photon must loose energy to the gravitational wave the photon makes when it travels through space, I assume? But I have never seen a calculation of the red shift due to this effect by mainstream astronomers. But prof. Jian-Miin Liu (Nanking University) has some interesting papers on this subject on the internet, using the GTR as his theoretical background. I would be very glad to hear of your views on this.
Not everything that has a gravitational field emits gravitational waves. Gravitational waves are emitted if the gravitational field is time dependent and has a time dependent quadrupole moment.
The gravitational field of a propagating photon, or indeed any propagating particle, does not have a time dependent quadrupole moment so it does not lose any energy by emission of gravitational waves.