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The idea that the moon has an affect on fishing stems from that fact that the moon causes minor tidal changes in the ocean. However, research conducted on how this effects fishing catch rates disproves any significant differences during different moon phases. On freshwater the moon effect is nil. That being said, since it is certainly brighter during a full moon, perhaps some fish may take advantage of the light at night and move or even feed more. But in the big picture I would think it does little overall. Under a foot a ice, probably even less.
Try telling a hardcore musky angler that moon phase doesn't matter.