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Former C-130 Flight engineer here, it was my job to calculate performance data for takeoff and landing.
Rain only matters because it increases stopping distance. It doesn't affect engine performance, and it never even occurred to me that a wet airplane would be heavier than a dry one. To get a sense of weights we care about, our empty weight is ~90k pounds and our max (peacetime) takeoff weight is 155k pounds. Performance numbers are good for 5000 pounds, so even if the water weighs several hundred pounds we'd never notice.
The most important factor in engine performance is the density of the air, which is driven by temperature and altitude. You get more power on cold days and at low altitude, less on hot days and high altitudes. (Which is why Denver has long runways)
There is a decision tree when planning a takeoff, and extended stopping distances due to a wet runway sometimes pushes you to use a higher power setting which is a bit less efficient. So the answer to your question is maybe a little, sometimes, but not in the way you think.
Rainy days will lower barometric pressure, so perhaps there's a performance drop, but not by virtue of the water on the plane?
We do account for barometric pressure, but it's generally a few tens of feet. So yeah, a teeny tiny bit, though you can have low barometric pressure without rain.
Plus any raindrops being sent through the engine will weigh more than normal air
Sorry I wasn't clear, it's the density of oxygen in the air. Rain will reduce that number, but by an immeasurably small amount.