x is actually the generated image they are testing against x0. A lower E(x; x0) means an image which fits well towards the objective based on the original image (depends on the task). The paper gives some examples. For example, for the task of image denoising, E(x; x0) is just the squared distance of the generated (denoised candidate) image x to the pixels to the original image x0. Obviously you would want this to be low in the generated version since it should still look close to x0.

R(x) is a regularization term to avoid overfitting. For example, in the denoising example, it could be a measure of the variation in color of x. Clearly, just taking x = x0, the squared error (E(x; x0)) is 0, but it will have high R(x) because of all the noise. That's why they try to minimize both quantities combined, so we get min_x E(x; x0) + R(x), to get close to the objective but also not overfit.

https://en.wikipedia.org/wiki/Regularization_(mathematics)

You're right, and here, they're including the prior R(x) in the model architecture. So R(x) = 0 here,

and they pose x = f_theta(z), where f_theta is a neural net.

So they can avoid overfitting by stopping the training at the right moment.

So they're not really trying to minimize E(x, x0), but rather to get a small enough value, so that they obtain good results. A too small value means overfitting.

I think x_0 refers to the corrupted image and x is the (denoised/upscaled/inpainted) image the optimization algorithm tries to come up. Without R(x) the optimal solution is simply x=x0. So we need a good R(x) that can measure how good or natural the solution image is. In this work the R(x) is constraining the valid x to those that can be represented by a CNN.