Does the moon have any influence over the ongoing cycle of food production in nature?
My first impression would be to say no, but the moon probably does have an influence to some degree over food production here on the Earth. It is a well-known fact the gravitational attraction of the moon pulling on the Earth produces tides, which are daily changes in the level of the ocean's surface. That would be the first factor that would affect food production, the daily fluctuation of the tides in the oceans, going in and going out.
Another way the moon could influence the food production here on Earth is the presence of the full moon, when the moon is half-way through the cycle of moon phases. I have personally witnessed full moons that cast enough light so as to emulate daytime visibility. That would be bound to encourage nocturnal hunters in their pursuit of food. A large, fall, orange moon is called a "Harvest Moon", dating back to times when farmers worked small farms harvesting crops from when they "could see" to when they "couldn't see". So it would be fair to say the moon does exert some degree of influence over the production of food here on the Earth.