Agent's home site

[Annatar]

Can I reveal Baduila (home sites: Goblin Gate, Mount Gundabad) If Goblin Gate and/or Mount Gundabad are in my discard pile? A revealed agent in his home site also needs a site card?

[Bandobras Took]

If one of your agents is revealed before it has moved, you must immediately choose which home site it is at - place the appropriate site card with the agent. If you do not have such a site card in your location deck, discard the agent at the end of the current turn.

You can reveal an agent at a home site if the site is unavailable. Keep in mind that they will be discarded at the end of the current turn if you do that.

[Annatar]

Thanks, as always Brandobras Took.

What if the agent moves but all his home sites are in my discard pile? For example, when I reveal Baduila, he has moved to Cameth Brin and then to Bree, but Goblin Gate is in my discard pile. Is it considered a legal movement?

From rules:

An agent is played face down and untapped. The
agent is at one of his home sites (no card required)

[Theo]

If one of your agents is revealed after is has moved, the first site card that was played must be one of its home sites or it must be in the same region as one of the agent's home sites or ir must be in a region adjacent to a region that contains one of the agent's home sites.

That is, (under the second two possibilities) the home site does not need to be included in the stack of agent's sites for the movement to be legal. The home site being in your discard is fine.