That's an interesting question. I think the answer partly depends on your climate. Here in the wet Northwest, I'd say it's worth considering because the heavy rains will wash nutrients out of the soil. I'd suggest putting a layer of organic material on the soil before covering it to help keep the soil in good shape.

Yes, where are you? Are things wet because water sits on top of ice some inches down? Does it sit covered in snow for long?

Where I am (Midwest), clay soil can have a pretty narrow window between soaking wet from winter and soaking wet from spring rains followed by rock hard in the hot summer. If I'm lucky I can hit a moment when the moisture is just right for digging. If you can manage that tarp - putting it on right before rain and taking it off afterwards to let it dry out - it should help. But if it's too wet to begin with it won't dry much with the lid on.

I'm in southern Ohio and you hit the nail on the head. I am trying to find the "sweet spot" so I can get in the early garden. I am trying to guess if it's better to leave it on through the spring rains or keep it off and take my chances. It takes me a couple of exhausting hours to put it on or take it off. I would prefer to have it off on dry days but I'd get caught for sure when showers pop up. It only takes a little rain to soak my well drained clay loam.

Placing an impervious cover over the soil will not allow the soil to dry out and will in fact hold moisture.

Moisture leaves soils by both drainage and evaporation and the plastic cover, or even a canvas, tarp can inhibit that.

I'd say if you're having to spend that much time on it (must be a big garden!) it's probably not worth it. If you just wanted to plant some early crops, how about doing this experiment with a small patch that is more manageable?