Siege of Winterfell allows you to only claim power for your house as a result of military challenges.

How does this interact with the prized mechanic, where you claim power when a prized character, attachment, or location leaves play, prized plot goes to the used pile, or when a prized event is played?

I would say that unless that card left play (or somehow that plot was cycled) during a military challenge, Siege could NOT claim the power.

That is correct. It's no different than being able to claim opposed power for MIL challenges (because you do so during a MIL challenge) but not for INT or POW.

Right. Thought so, but I wanted some clarification.