The Windows paging file is the file where Windows writes all the RAM whose processes aren't using. Basically is you have two programs which require a lot of RAM, it makes no sense to have both programs loaded onto RAM - either if you don't have enough RAM or some other case. Thus, there's this "Virtual RAM" - the paging file. So Program 1 would run off RAM, and program 2's information would be stored in the paging file. When you switch programs, the RAM and the paging file would swap - sometimes why it takes Windows a few seconds to switch to another application.
The Photoshop scratch file is basically the same thing, it stores information which don't need to be permanently kept. Such files include "History" of the document, so you can undo changes.
So if you do have another HDD, it would make sense to keep both on two different hard drives, but as mentioned, for causal usage it wouldn't really effect usage.