The inner garden of the Imperial palace can be visited, free of charge, twice a day, with a guide.
You need to select the day and time, fill the application which can be found here, and send it by mail. You will then receive a confirmation.
If you have already seen the Kyoto palace, then this visit might be a little disappointing. After a 20mns video, you will taken for a walk around a very limited area of the Palace grounds, spending most of the time on a tarmac pathway and not seeing much, as shown on the pictures below...
As long as you don't have too high expectations, it is a pleasant -and free- way to spend a couple of hours in an uncrowded place...
Poras Raoul ポラス ラウル