I don't know how much the moon has to do with the spawn, but I do think it influences it. In April and May perhaps they spawn when the water conditions allow since there is so much variation in temperature, clarity, and water levels in the spring. But I have reliably caught fish bedding around the full moon in June each of the past 3 years. Just hit it randomly a few years back and have intentionally targeted the June full moon since whenever I can. Never used to fish around beds in June because I believed the spawn was long over by then.

Having caught so many over the years on beds in late April and through May regardless of the moon phase I think the early spawn may be less connected to the moon phase. May just be me trying to justify my experience though, since in recent years I'm not on the water all that much due to the high level of church and school activities the kids are involved in during this time of year.