1. The problem is that the 1 TB HDD contains the 500 MB System partition that's needed for a viable boot to the OS (which I assume to be Win 10, yes?). (Whenever you pose a query along similar lines always indicate the OS you're working with.)
2. So when you currently boot to your 120 GB drive the 1 TB HDD should be connected. Generally if it's not, the system will not boot. Win 10 is a bit more forgiving; at times it will boot to the OS under these circumstances, generally after a reboot. But don't count on it.
3. It's likely this problem arose because you originally cloned the contents of the then-1 TB-boot drive to the 120 GB SSD but you did NOT disconnect the 1 TB HDD immediately following the disk-cloning operation and initially boot to the newly-cloned SSD as the ONLY DRIVE CONNECTED at the time. Are you following me?
As a general proposition it's good practice that IMMEDIATELY following a successful disk-cloning operation the source drive be disconnected from the system and the initial boot to the OS be undertaken with ONLY the newly-cloned drive present in the system. Capiche?
4. Frankly, given your present configuration, probably the most practical course of action would be to fresh-install the Win 10 OS onto the 500 GB SSD in order that drive would become your new boot drive. Now of course that would mean you would have to manually install any programs currently on your 120 GB boot drive and copy/move your personal data, audio/visual files, etc. over to the 500 GB drive. So that would be something of a chore.
Is that a viable option for you? If so, you may want to pursue that option.
5. On the other hand there's another possible option...actually a more simplified process, but there's a chance that it will not be successful. I'll go into some detail after explaining the potential process.
A. After disconnecting the 500 GB SSD from the system, boot with only the 120 GB SSD and the 1 TB HDD connected.
B. Presumably the system boots to the C partition on the 120 GB SSD. At a Command Prompt (Admin) type: bcdboot c:\windows /s c: <Enter>
C. System returns "Boot files successfully created."
D. Shut down system and disconnect the 1 TB HDD drive from the system.
E. Boot the system with only the 120 GB SSD connected and determine if you get a viable boot to the OS and the SSD functions problem-free.
F. If so, it worked.
Now for the caveat...Understand there will be a chance that the preceding process might not work and you'll be left with an unbootable SSD. The likelihood is it will work but the possibility exists that it might not. I want you to be aware of that.
It's the reason that when we work on other users' PCs involving this or similar types of reconfigurations, we FIRST clone the contents of the drive we're about to manipulate to another drive so that in the event of a failed operation we have the wherewithal (the clone) to return the system to its previous state. Are you with me?
6. Anyway...in the event of success, you can reinstall the 500 GB SSD and clone the contents of 120 GB drive to the larger drive. I assume you have a favorite disk-cloning program in mind. And don't forget to DISCONNECT the source drive (120 GB SSD) IMMEDIATELY following a successful disk-cloning operation and boot ONLY TO THE NEWLY-CLONED 500 GB drive that's connected in the system. After you determine it boots & functions problem-free only then should you reconnect the now secondary drives. Capiche?
Let us know how it all works out, OK?