I would fuse a small piece of the same (or similar) fabric underneath the cut on the edge. Steam-a-Seam would be a good choice because it is very stable but does not change the feel of the fabric too much. SAS also allows you to reposition, so you can get the placement exactly right before you permanently fuse.
If anything still shows afterwards (and perhaps to protect the cut from fraying years down the line), I would place a small applique over the cut. Depending on the total design, you might be able to scatter small appliques in other places on the quilt so they look as if they were intended.
If you really do not want the appliques to show, do them in matching Kona cotton white so they are very subtle. You can just quilt over them, if you like, to further imbed them in the background. For a Dresden plate, I would probably make small flower shapes.
Oh, and I think the idea to put this block in a corner is great! You could decorate each corner block with a few of the appliques so it looks planned.