Images are only imported into Lightroom once. That import process points Lightroom to where they are stored in your system. After that, it is simply a process of moving images when necessary. The best thing to do is to import once, and then leave the images in that location. But in your situation it sounds as if moving is necessary to enable you to accommodate more images.
There are two different methods of moving images in Lightroom. Since they are already imported and available in the catalog, you cannot use the import dialog. Instead, you can go to the library module and locate the folder that you want to move in the folder tree in the left-hand panel and drag it to the other drive if it is available in the library folder tree. I think you will find this is the recommended way in much of the Lightroom literature that's available. However, it seems that when using this method some images for some reason don't get moved. So there is another method that I personally prefer.
My preferred method is to use Windows Explorer or Finder to move images to a new location. When doing this, I suggest that you move complete folders or even folder trees if possible. When you open Lightroom after moving those folders Lightroom is going to indicate that the folders are missing. But not to panic. All that is necessary is to right-click on the topmost folder that Lightroom indicates is missing, and choose the option to find missing folders. Then you will be presented with a dialog that will enable you to browse to the new disk drive and locate the folder and choose that location. Once you have chosen the new location everything will be synchronized and your work can continue.
As I indicated previously, it's really best if you don't have to ever move images. But if it becomes absolutely necessary it's relatively simple to do. If you only want to move a few images from within a folder it's possible to highlight the images in the library grid view and drag/drop them to a different folder in the folder tree. But if you're going to do an extensive move my preferred method is probably more reliable. Here is a page of instructions that might help you understand the process of finding missing files or folders, Here are the instructions on how to get things synchronized again:
Many thanks Jim, that worked for me.
The most important learning I gained is that the file/folder tree in the new (or backup) location must be identical to the structure that the Catalog is built on.
If and when I can summon up the courage, I'll try a recovery exercise by deleting my catalog and then importing it from backup, and trying to get it to work on my backup storage. Maybe I'll get back to you again .