I would think the you would want Photoshop to use F: as its scratch disk because Photoshop can need to use a large amount of Scratch space. You would not want Photoshop filling up you C: SSD disk. Windows needs free space on you C: disk to run well. As you know Photoshop also creates temp files. The space required for Temp files normally is much less then the space required than Scratch space Photoshop can require. Only is you creating huge Image the are required to be save as PSB files. Auto recovery space requirement would the be large. If you can use Photoshop startup shortcut CTRL+Alt to configure F: as Phtoshop Scratch disk the set F: as the onlt Scratch disk. Let Photoshop store its temp files on you C: in your user ID Temp space.
I the only thing I can think of why you can not see F: in your Photoshop Preferences is somehow F: is listed by you Windows 10 system as removable media. Photoshop will not list removable media as usable for Scratch space.
No. F partition and disc is visible in system as full internal storage same as other discs. For more, any external discs connected to system are also vsible in Photoshop preferences. And i don't think in any way this can be disc problem, becouse any other application have no problem using F partition. Still, only PS does't see it in scratch disc preferences (but when saving files it is vidible in system normally).
And now, most funny part. I've installed PS on F: partition. Completley normal, withowut any rpoblems, PS also works fine, but still can't see F disc for scratch disc...