I see some city lights in North Africa, North Australia, and across Indochina. I barely see any city lights in North America, Europe, Russia, and China.
I'm not seeing that in my game. Are you looking at the dark side of Earth? The lights won't show on the bright side of the planet. There also may be clouds blocking the sight.
Also, the amount of city lights depends on how close the planet is to max population. If you only have three or four billion people there (out of 16b), you won't see many lights.
Is there any way to fix it
I'm not familiar with the graphical side of modding the game. However, the file governing the look of the city lights seems to be the Light_Noise.png in the GalCiv 2\DarkAvatar\Gfx\planets folder. Still, I'm not sure how much you can do with it.
Also is there a way to fix it in my current game so I don't have to restart it just to have graphics update and be fixed?
The game stores most of the data in the savegame to prevent patches from braking the saves. There are only a few exceptions to this, like the dialogue between the races. Planet graphics are not one of them, as far as I know.