It regains normal influence once you hit 'Turn". Don't know why, but the true influence is only restored after the first turn after you reload a game.

I explored this over the weekend and had some surprises.

I noted several enemy planets with the skull and crossbones, then saved the game and reloaded.  The planets did not appear with the symbol, as expected.

Next, I attacked the planets' defenders.  After each attack, the symbol reappeared.

Next, I reloaded the saved game again.  I had the highest espionage level on that race, and just toggeld the "View" option for each of those same enemy planets.  In each case, the symbol reappeared after I toggeld "View."

So, it would seem that if a player wants to preserve the chance of an Influence "flip," it might be best to toggle each enemy planet, or other wise make the game engine "consider" the planets before hitting the turn button.

In some cases it works to your advantage, in others not.

It all depends on which side of the scale your base influence falls.