Your pictures are not stored in the Photos app-- that's the instructions for how to work with them. The pictures are stored in the Photos Library, probably called Photos Library.photoslibrary, located in the Pictures folder.
If you have dragged the Photos app to your new hard drive-- trash it! It shouldn't be on the hard drive. You should still have the Photos app in your application folder where it belongs.
So, drag the Photos Library from the Pictures folder to the new drive. I would rename it, like NewPhotos, or something. Then double click NewPhotos. It should open Photos and you should be able to look through it. Look at Imports to see if some of the new pictures are there. Drag one new picture of anything from anywhere onto the NewPhotos Library icon that's now on your hard drive, and make sure it shows up in Photos-- we should make absolutely sure that Photos is looking at the new library.
In fact, go to Photos>Settings>General and look at the path for the Library:

I had renamed my library to remind me what it was for. You can see that mine is in my Pictures folder. Yours should show that your Library is on your new drive. If that's good, tap the Use as System Photo Library. Mine is grayed out, because it is already the System Library.
If you're satisfied that all is well, then go to Settings>iCloud

and turn on iCloud. Make sure everything looks good before deleting the Library on your Mac! In fact, you could again drag that Library to the new drive as a backup, since that one has a new name and is no longer the System Library.
You can see all your Libraries by closing Photos and then option-clicking on the Photos icon. You'll get a dialog like this:
It shows the path to each one and identifies you System Library. The System Library is the only one that connects to iCloud.
Let us know how it goes!