What is the state-of-the-art on using computers to "clean-up" images?
The closest thing I know of to the "computer: enhance!" trope in real life is the family of Single Image Super-Resolution techniques. That page shows a number of examples of the results on various images. You can see that while it improves the visual quality of the enlarged images, it's a long way from what you see on TV where they can read the text on a letter that's reflected in a wine glass, or recognize the face of someone standing in the shadows, etc.

The technique basically works by observing that a single image often contains recurring instances of the same patterns or structures. When you have multiple copies of the same pattern, they often occur at different sub-pixel offsets, i.e. aligned differently to the pixel grid, which means that each copy contains slightly different information about the underlying pattern, and by putting them together you can recover (really, guess at) a higher-resolution version of that pattern. This can then be used to "fill in the missing detail" wherever that pattern occurs. This is a very handwavy explanation, but you can see the paper (linked from the above site) for more details. Most of it is automated, but I'd guess it still requires a fair amount of human parameter-tuning to get good results.

The paper is from 2009, and it looks like there have been a few follow-ups since then, but only incremental improvements.

Also note that if you have a specific use case such as reading vehicle license plates, I'd guess it's possible to use machine-learning techniques to get much better results than you can in general. In that case, you have a concrete, known set of possible shapes (the characters that can appear on a license plate, in the particular font used for license plates), and rather than "enhancing" an image of literally anything, you're just trying to find the set of characters that best matches the image you're looking at. That's not my area of expertise, though.