I wouldn't way it's the Moon the 'culprit' but rather the interaction between the sun light, passing by the Moon's surface, and, the most important, our atmosphere which is highly charged with suspended particles which diffract light and that make the sky polluted by light. That's precisely why it's blue during daylight and close to black at night time.
You can check this, this and this out for more detailed information.
The same phenomenons apply to the Moon light I guess.
And that's why the higher terrestrial optical instruments are placed (I'm mainly thinking of large telescopes, but this obviously applies to our digital cameras) in relation to the ground, the better the images.
Of course, this is valid if you make abstraction of any other sources of light such as urban lights.
That said, one of the best way to get rid of "random pollution" is to go for image stacking.
There is many many softwares out there to post-process astronomical images...
One that I like is DeepSkyStacker.
But as the first comment hereunder by scottbb says, indeed, is that you won't get rid of the part of the pollution which is based on the atmosphere physics and how light actually travels in this medium, which can be coarsely modeled because of the well known vertical composition of the atmosphere and hence, the global variations of the refraction coefficient for example.
The stacking will only get rid of stochastic or not predictable (and mostly small) perturbations, either the electromagnetic ones at the sensor level, or the environmental ones such as the atmospheric turbulence that affect astronomical observations, which also are the main cause of star twinkling and which led to the invention of adaptive optics for example. But I guess we won't have such nice technologies in our digital camera during our lifetime.
Finally and most of all, If you have to take away one thing; avoid the Moon, especially if the atmosphere is not perfectly clean such as after a large rainy meteorological event (because of the light diffraction). It is statistically worth waiting an average half a month to get good shots.