First of all, you should realize that the SSD did NOT "assign itself as C:". Windows does this. Windows ALWAYS assigns the C: label to whatever device it just booted from.
You post says you "cloned the OS from the HHD onto the SSD". Does that mean that you did NOT clone anything else from the HDD onto the SSD? So, all of your applications software is still on the HDD, and that has become the D: drive or something like that? Well, that's the source of the problem. What you did clone to the SSD includes the Windows Registry files, which contain all the details of every application software package you have installed. And of course, those details include the locations of all those app files, and that location in every case is on the C: drive. Except that now they are not there - they are on the HDD still, newly renamed to D: or something. So every time you try to launch an app, Windows looks for it on the (new) C: drive and can't find it there.
How to fix? Not sure at all - never had to do this. Maybe you have to re-Install all your software on the D: (or whatever) drive. Maybe you could edit all the Registry entries, but I REALLY don't recommend that unless you do thorough backups of the Registry first and know exactly what you're doing.