I don't think that's clear at all! If the on-play effect of Clairvoyant is "+2 cards when somebody plays the set-aside card", playing Clairvoyant twice should cause that effect to happen twice.
Let's step through it, assuming "until then":
1. You play Throne Room, choosing Clairvoyant.
2. You play Clairvoyant, choosing Woodcutter. You set aside a Woodcutter from the Supply.
3. You play Clairvoyant again, choosing Woodcutter again. You set aside another Woodcutter from the Supply.
4. Someone else plays Woodcutter. You check your Clairvoyant-set-aside cards, see that there are 2 Woodcutters there, and you get +2 Cards twice -- i.e., +4 Cards in total.
5. Your turn comes again. You return Woodcutter to the Supply, as instructed by Clairvoyant when you played it. You return the other Woodcutter to the Supply, as instructed by Clairvoyant when you played it again.
6. Clairvoyant is still in play, but it is no longer doing anything. It will be discarded at the end of your turn, and so will Throne Room.
Again, I think the easiest way to think about Clairvoyant is to compare it with Haven: it moves a card immediately, and moves it again at the start of the next turn. It just so happens that Clairvoyant does something else inbetween these two events.
Here's my issue with "while this is in play":
1. You play Clairvoyant, choosing Woodcutter. You set aside a Woodcutter from the Supply.
2. Someone else plays Woodcutter. You check your set aside cards, see that Woodcutter is there, and you get +2 Cards.
3. Your turn comes again. You return Woodcutter to the Supply, as instructed by Clairvoyant when you played it.
4. Clairvoyant is still in play, and its while-in-play effect is still active. It will be discarded at the end of your turn.
My issues are with steps 2 and 4.
- In step 2, the "while this is in play" clause has to use a tortured wording to refer to those cards that were set aside by Clairvoyant. Either that, or introduce a Clairvoyant mat.
- In step 4, there is a somewhat tricky game state where Clairvoyant is in play but Woodcutter is no longer set aside. I can't think of any way to exploit this ATM, but it seems more elegant to close the loophole.