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What side would you prefer to play as?

Definitely white. The simple reason is that black's king will find no comfort this game. Actually even strategically, I'm not sure how black responds to something like a4. It looks like we're just winning the c4 square by force.
#1 I do not see how white lost pawn h2.

With the pawn on h2 it looks equal.
Example:


On 4 c4 black just plays 4...g6, equal again.
Example:


#2 on 4 a4 black can just push 4...b4, equal again.
Example:


@tpr In this sort of position you cannot just ignore the missing h pawn. This is a critical change in the position from the games you're referencing.

Black's play in these position usually relies heavily on the fact that white's play is somewhat more difficult to get. But here, white has immediate play and black's king will never be safe anywhere on this board. So for instance in that game after 4. a4 b4 5. Bg5 Ne4 we already have ideas like 6. Qd3 when black looks to be in some trouble. After 6. .. Nxg5 7. Nxg5 seems like black's position is already collapsing.

Though I don't think Bg5 is the move there. Instead 5. Nd2 is really tempting since taking that pawn looks mighty dangerous after either of Nc4 or e4.
Half open h file for white along with probable kingside castling for black leads to some easy to understand attacking ideas for white in the middlegame.

I'd pick white, because the way forward seems more clear.
#6 but as people have said the absence of the h2 pawn is a plus for white, I'm not sure it does. If white has to open up the h file, that would take more time, so any analysis wouldn't translate to a real position.
If white pushes to h5 and lets the knight take it without recapturing, we would question that line.

And if white can't win the pawn back (or checkmate), black's got to be at least the favourite in most endgames, so I'd take my chances with black.

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