Is Mojave no longer (or never was) compatible with the game?

I recently tried running the game again and all I see is the icon come up in the Dock and then just disappear.
No error message, nothing.
When I try running it from the Applications folder, I get 'The application “Master of Orion” can’t be opened.'

I did the File Access in the Security Tab, Verified the files, even did First Aid on my drive, but the game won't run.

What kind of wrapper does the game use? Can it be upgraded? Or edited?

HELP

Both of my Macs are running 10.15 now, but the game worked for me on Mojave in the past. If you haven't already, I'd suggest in order:

1) Right-click the app and choose open from the contextual menu (often necessary in order to get the option to open the app if Apple didn't recognize the developer).
2) Reinstall and try again. I've on rare occasions had the same problem for no clear reason and resolved it this way. Downloading via a different method might also be worth trying (GOG Galaxy if you used the standalone installer before, or vice versa).

Good luck.

I have an external eGPU connected to a second monitor but I don't know why that would change anything?
There is an option in INFO to 'Prefer External Graphics Card' but that should HELP in playing the game, right?

What other methods are there in downloading the game?
Thank you for your time.

You could use the standalone installer, either from GOG Galaxy or the website, or have GOG Galaxy install directly. I don't know that switching among these would make a difference but it's been suggested to me before by GOG support when game I downloaded via the website wasn't working.

When I've had the issue you described (with another application, not MoO), just downloading and reinstalling from the same place was enough to solve the problem, or else the application wouldn't work at all.