I think it would be hard-pressed to do it with one pawn. The difference between the best computer and the best human is very small by our standards, 1 pawn would make all the difference, just trade off pretty much.
Can Houdini beat Carlsen 3 pawns down

Houdini will stilll crush Carlsen.As long as the pawn removed wouldnt be a vital pawn in the king's protection like say an f pawn,but even then Carlsen will be fighting to draw.Take away the h pawn or a pawn or both the center pawns.That would be a nightmare for Carlsen.An open position and an accelerated development is a recipe for miniatures.

From the onset or later in the game?
If it is from the onset, then will the opening books have become worthless. Carlsen will chose openings in which the opponent will miss that particular pawn. For instance, when lacking the a-pawn, then is that a problem in a lot of e4 openings. The computer has to find his way without the help of opening books and engines are supposed to be worse in positions where understanding is more important then sheer calculation.
When the pawn is selected later in the game, then I think it depends partly on the selection method how to come to the choice which pawn is to be removed. It will make a difference if Carlsen can pick a pawn by his own choice or the chess engine can remove a pawn by his own choice or that the chess engine can present a number of pawns of which Carlsen is allowed to remove one of them for instance.
Houdini will stilll crush Carlsen.As long as the pawn removed wouldnt be a vital pawn in the king's protection like say an f pawn,but even then Carlsen will be fighting to draw.Take away the h pawn or a pawn or both the center pawns.That would be a nightmare for Carlsen.An open position and an accelerated development is a recipe for miniatures.
No, the difference isn't that big, Carlsen would never lose being three pawns up from the start.
GM Dzindzichashvili drew a pawn odds match against Rybka in 2007, and Dzindzi never reached a 2600 rating even at his peak in the 1970s as far as I know. Even if engines are much stronger today than Rybka was in 2007 there's no way a 2850+ player would be crushed miniature style with three pawn odds.
it depends solely on carlsen's ability in preventing the position from exploding tactically. I think Houdini can beat Carlsen in some games but he will win more games that Houdini does.
There is not much that can explode tactically when one player starts with a lost position and little to work with against such a strong opponent as Carlsen.
Supposing Houdini started with three pawns down then do you think it could beat Carlsen?