I found a fix, at least for my PC. On a Win7x64 system using 32-bit N++, I had the same problem with every post-7.6 version I had installed. None of them ever had the Plugins install correctly. I got it working, finally, with 7.8.9, but it’s likely the same issue as prior versions. I’m replying to this because it seems to have the most information/discussion.
The problem was that the GUP.exe file in the notepad++/updater/ folder wasn’t ever starting. I watched my Task Manager and checked windows error logs… it never started. So I had a hunch, anc ran the GUP.exe program directly, and Windows popped up its UAC nag window ('Open File - Security Warning). I unchecked “Always ask before opening this file” to clear it with the UAC, clicked ‘Run’, and it showed the standard Notepad++ Available Update message. I clicked ‘no’.
I then re-opened Notepad++, used the Plugins Admin to install a plugin, and it WORKED. Hopefully this will help others with the same problem, which was (for clarity) that selecting to install a plugin through the ‘Plugins Admin’, click ‘Yes’ to restart, and NPP closes but never re-opens on its own. When opened manually, nothing was installed.