Does the Moon capture radiation pressure from the sun causing momentum from photon propulsion? Could a planet's main use for natural satellites be that of a photon sail?
If we assume the 15 CM per year is accurate (which I'm not 100% sure it is), then it's possible to answer this.  In a loose sense, yes, the pressure should move the moon away from the sun faster, but more accurately and the real answer is that doesn't happen because there are 3 bodies in play and when you have 3 body gravitation, things get a lot more complicated.

First, just for fun, a source for the 15 CM.


  The moon away each year by photon momentum.The moon is the perfect
  color,size and The moon has a mass of 7.35 x 10²² kilograms. It is
  only about 60 percent as dense as Earth so it should be effected by
  the photons from the sun causing movement.And that movement should be
  combined with earths 15cm giving a new total distance a year right


If we only look at equal force per surface area, which is what you're describing, it's not density, it's the ratio surface area to mass.  The Moon's surface area is about 7.4% of Earth's (about 1/13.5) and it's mass about 1/81st of Earths.  Source.  81/13.5 = about 6.  That means given equal pressure, the moon should accelerate 6 times as much which corresponds over small distances to 6 times the distance or 90 CM per year - BUT, that's if you ONLY take into account the pressure.

The effect of the sun losing mass has equal effect on both the Earth and the smaller Moon.  

And a 3rd factor to consider is the tidal bulge on the sun caused by the Earth-Moon system, which is very small but all these effects are small.   The Earth-Moon system creates a tiny bulge on the surface of the Sun and because the Sun rotates ahead of the Earth-Moon, that tiny tidal bulge has a tiny pull on the Earth-Moon system that slowly accelerates them and pushes them slowly away from the sun.  That effect is equal for both the Moon and Earth too.   

All 3 of those are factors in the 15 CM per year estimate and only one of them has 6 times the effect on the Moon than the Earth.   The planets Venus and Jupiter might also be factors in that 15 CM per year estimate too, but lets leave that alone for now.

If we consider the 3 body problem it gets very mathy, but I'll just talk about how it applies to solar pressure.   Lets start with a picture.  



When the Moon is waning its moving towards the sun and any pressure from the sun slows the moon down, (a tiny bit) and that slow-down moves the Moon closer to the Earth which speeds it up even more - funny how that works, pushing to slow something down in orbit and it goes faster - but that's how it works, cause potential energy converts to kinetic as the orbit drops, kinda like how falling makes things go faster.

When the Moon is waxing, it's moving away from the sun and any pressure from the sun speeds it up which moves it away from the earth and that in turn, slows it down.  So that's your answer in a nutshell.  The solar pressure doesn't push the moon away from the sun so much as it pushes the moon into a lower and then higher orbit around the Earth depending on where the Moon is in it's orbit around the Earth and the various positions in both the Moon's and Earth's elliptical orbits.  The overall effect is very difficult to calculate and I suspect it could go either way depending in part on the timing of large coronal mass ejections hitting the moon and in part on any possible resonance between the eccentricity of the Moon around the Earth and the Earth around the sun, er, I think.

What is safe to say is that the Earth-Moon system obeys the same surface area to mass ratio the Earth alone because that's simple momentum which needs to be conserved.    The Earth-Moon system has a combined surface area of 1.074 Earths and a combined mass of 1.012 Earths, so the Moon being in the Earth-Moon system makes the Earth move away due to solar pressure about 6% faster.  15 CM to 15.9 CM.  Not a very big change, and it's possible that the 15 CM estimate is based on the Earth-Moon barycenter not the Earth itself.

So, the effect is more of a wobble in the Moon's orbit around the earth than an actual force that pushes the Moon away from the sun faster than it pushes the Earth away.

All this, is, of-course, unimportant compared to the much bigger solar tidal force on the Earth-Moon system.   As the Moon gets closer to the sun (new moon in the picture above) The sun effectively pulls the Moon a little bit away from the Earth and when it's further from the sun (full moon), The Moon is effectively pulled towards the earth.  The net effect of this solar tide is a measurable wobble in the Moon's orbit around the earth.  The effect on the Moon's orbit around the earth due to photon pressure and coronal mass ejection pressure - basically insignificant.

Hope that's clear.