No, the cycles of the moon do not affect water evaporation. The moon affects where the water is on the Earth through gravity (tides), but the moon does not, such that we know, affect evaporation, clouds, or rainfall.

I don't believe the cycles of the moon affect evaporation over the ocean. The main variable that is affecting this evaporation is how hot it is, or how much sunlight there is in a particular region. For example, that's why at the equator, where sunlight is more intense, there is more evaporation from the ocean than at the poles.