when precipitation goes up does evaporation increase?

It depends on the weather conditions after the precipitation. If it precipitates and then the sun comes out and produces latent heat on the ground surface, the water that precipitated will evaporate and cool the ground and air on its way up. This is another reason why deserts are so hot. Because there is little to no precipitation, there is little to no evaporation of water to cool the ground or air.

If It is a cold environment and it snows, There is also not as much evaporation as a warm watery environment. Water must be in liquid form and be exposed to ample latent heat before it can evaporate off the surface. For snow precipitation to occur the temperatures must be at least 32 degrees F. If it is cold enough on the surface, the snow will remain in solid form, preventing evaporation.