Unable to save edits in Photos on MacBook Pro

I suspect that the Mac is running out of storage.

Could you please select the icon of the internal disk (Macintosh HD) in the Finder and open the Info for this disk with the keyboard shortcut ⌘i ?

In the Info panel look at the "General" section - you may need to click the ">" to the left of "General" to see the details. What does it say how much storage is used and what is the capacity?

This will tell you, how much storage is currently free.

When you are saving photos in Photos, the Photos.app is storing your photos not in the app, but in a package called Photos Library.photoslibrary.

To see, where your Photos Library currently is, open Photos, if it is not yet running.

Then go to the main menu bar in Photos, click the menu entry "Photos", then "Preferences" (or "Settings", if you are using the most recent version of Photos.

In the Preferences or Settings panel click the "General" tab, if it is not yet open. At the top of the panel is a button "Show in Finder" and the path to the Library. Use the "Show in Finder" button to open the folder with your Photos Library.

Where is this folder and what is it called? Is it the Pictures folder in your user Home folder or somewhere else?

Even though your Library is on an external drive, Photos uses the internal drive for its operations. The usual recommendation is that 15% of the hard drive storage should be kept free-- that's 75 GB on a 500 GB drive. I have a 1 TB drive on a Mac Air, and I notice slowdowns when I have less than 20% free.

It looks like you have only 35 GB free, only 7%! As léonie suggested, this could be your problem. You should see if you can move some of the storage on your Mac to the external drive to free up some room.

Could you tell us more?

What kind of edits are you doing in Photos?

Are the photos in an iCloud Photos Library or a Shared iCloud Photos Library? If you are using a Shred iCloud Photos Library, are you the owner of the edited photos?

Where is your Photos Library stored? In your Pictures folder or on an external volume? And how much free storage do you have?