No where to add new scratch disk in Photoshop (2023)

And the kind of drive may also make a difference (see »Supported drive formats for scratch disks«): 

https://helpx.adobe.com/photoshop/using/scratch-disks-preferences.html

As others have mentioned, you need another physical drive attached to your computer. Do you have one attached?

The scratch disk is not a Photoshop setting. It is actual, physical disk space. You need to purchase a new/bigger drive, or clear out space in your old.

What kind of hardware are you talking about? 

Photoshop should automatically detect available disks, and then you will have the option to select which disk to use as the scratch disk. I am not aware of a button to “add” a disk, as this should be done automatically. Make sure that your disk is properly connected.