Well...It looks like I've answered my own questions. It seems like what I needed to know regarding this annoying problem, and how to correct it, could be found at the HP Help website:
HP Notebook PCs*-* Low Disk Space Error Appears During or After Backup*
It describes step-by-step how to go in and delete a couple files in the D Drive's Recovery Partition. And then how to set it up for all future back-up files to be saved to a disk using your DVD Drive, rather than prompting you to save it to the D drive. Having it set to save your back-up information to the small D drive partition is what makes your PC display the message telling you there's not enough room left on that Drive. And believe me...you'll be getting that message all the time...not just when its time to do a back-up. So now since I've just completed the procedure described at this help site...I'm hoping this'll end my problem once and for all. If it doesn't, you might see me back in here again asking you guys to help me out. I hope this helps any of you that find yourselves with a similar problem. Thanks.