justmoth: I have this issue too, and I managed to brute force a workaround (without having to reinstall GOG Galaxy).

1. Open the page for the game with the broken cover art (so you see the game background, screen shots , ect)
2. Click the slider icon at the top of the page and select Edit.
3. Select Media (Should show the Title image space with the “Could not load the image” message)

Now here's where it gets tricky...

4. Open the GOG Galaxy webcache directory ( Unusually C:\ProgramData\GOG.com\Galaxy\webcache\[Big old number])
5. Select the directory of the platform that the messed up game is from (for example Steam)
6. Locate the App ID for the game (may need to trial and error to find it, or if it's a Steam game go to the store page of the game and the App ID will be the number in the URL. For example, 447150 for the game Ape Out)
7. Delete the App ID directory for the game.
8. Alt+Tab back to the Edit Media widow of the game in GOG Galaxy.
9. After a moment or two the “Could not load the image” message is replaced with a broken image icon and you can now add a new image. (if the message doesn't go away after a while try Alt+Tabbing the Game page and then back to the Edit Widow).

The timing of this is finicky and if you navigate way from the Edit window when the broken image icon is present without adding a new image (or it just doesn't work) you'll have to delete the game directory again and retry.

There may be easier ways of fixing this, but this is the only thing I figured out so far.