lousy mirror corrected by software Would it be possible to get a terribly shaped mirror, like a large sheet of something reflective, give it a general shape like a telescope mirror, but not anywhere near as precisely shaped as in a proper mirror, then use software to correct the image collected by the CCD into likely images?

The software could "learn" by being pointed at various objects , sun, moon, planets, stars, houses, etc, to learn it's own mirror errors, which you could correct, with the correction data saved as a filter,,much like facial and object recognition software operates, then use X number of high end graphics cards to correct the image in real time.
 Seems as though the software correcting for the terribly made mirror would eventually create a filter that would give a good image. Even if the errors required a lot of storage space on drives to create the filter, it should be able to correct the images over time into something recognizable and maybe accurate.

Another thing would be editing software for the image , you could manually or automatically bend the observed objects lines, correction software would straighten or curve lines in initial images, and the correction you apply would be applied to your terrible mirror's error correction filter.   Your terrible mirror could be an old silvered Pyrex plate, or a large piece of Mylar balloon material mounted on a shaped frame, or an intentionally formed piece of sheet metal, or such common things with an inaccurate mirror shape that's not ground to a wavelength of light as current mirrors are. Information not received and reflected by the lousy mirror could be filled in by the   software after it learns for a while.  The cheaper but much larger mirror might fill in some of that lost information~~~
There are a couple different ways this could go, depending on where in the optical train (of other powered items such as mirrors and lenses) this particular mirror is.   

In a simple case, such as when you view a "funhouse mirror," the mirror causes pure distortion, and not any defocus or aberrations.  This is easily corrected either with a mirror of opposite shape, or by remapping the image based on the distortion parameters.  Remember: distortion does NOT blur anything.  

If the mirror is in an intermediat focus plane, then you essentially have created the heart of an adaptive optic system -- albeit your mirror is not "helping" .  In this case, the mirror's shape is causing the optical wavefront (phase) to get degraded. The phase delay, which is a function of optical path length, varies across the mirror.  In this case, you will need some sort of phase sensing device, such as a Shack-Hartmann sensor, to identify the errors and possibly drive correction systems.  

I suspect your overall question is more closely related to the first of these cases, so a simple remapping will suffice.  The typical approach taken by us optickers is to image a large picture consisting of dots on a grid.  We expect the image dots to be on a rectangular grid, so when they aren't we know how much to "move" each pixel in the camera to get the correct image.