I can tell you historically I have been totally convinced coyotes have been conditioned/educated towards less movement on full moon nights in my state. Traditional moonlight hunting in my state has continued to get tougher and tougher which had led me to this conclusion. It was this extra hunting pressure, I was attributing to the coyotes changing their behaviors.
I did track this data the last few years. There are so many other factors including cloud cover, barometric pressure, etc that it is hard to say it is totally dictated by phase of moon. My data has been inconclusive. Last year, I had two of my worst hunting nights of the season during a full moon phase, so felt like I was confirming this hypothesis. However, as snow conditions changed, I hunted a lot more towards the end of the winter season. The last 2 full moon phases of our night season, I had very good success.
My gut tells me there is more predator activity right before the full moon rises and after it sets on full moon nights. However, I have killed lots of coyotes on bright nights and if you can go on a full moon night, why not.