Mysterious moves without apparent purpose can be good. But they could just be without purpose. Why move a pawn on h7 to h6, when it’s entirely possible that later you will wish it were still on h7 and it can never go back?
Hm, we do famously have “mysterious rook moves” which usually means moving a rook behind your own pawn that you’re going to push, rather than to an already open file. But we don’t have “mysterious pawn moves”. Amazingly often, virtually always, they are a bad idea if not well thought out, and a great example is from todays game Anand - Gelfand in the World Championship match.
On move 23 all the commentators assumed Anand would play b3 to push Gelfand’s knight back from a4 and make b2 available for his king. But Anand played d6, granted a pawn move but nothing mysterious about it in that he was advancing the passed pawn. None of the commentators expected it. Even after the move, for several minutes Jan Timman ASSUMED that Anand had played 23. b3, Nb6 24. d6, that was the position he discussed. Also the Russian live commentators continued talking as if 23. b3, Nb6 had been played first. Sergei Shipov safely posted no meaningful comment about the move at all and waited to find out if Anand had really made a terrible blunder. Gelfand played the fairly obvious 23 … Rfc5 threatening mate. (23…Nb6 would have probably been easier) Didn’t Anand make a mistake?
No.
Anand played 24. Rd1! and the strength of the position with UNMOVED pawns in front of White’s king was apparent. Black has a lot of pieces swarming but no way in. It emerged later in the press conference that Gelfand should have played 24 … Nb6!! 25. Nc3, Rd5! (must be a computer to see that) and then he would have gotten his draw with less worry.
Don’t move pawns in front of the king without a reason. If Anand is teaching that lesson today to grandmasters, we fish can do well to respect it too.
Just generally, don’t move a3 or h3 unless you have a particular need to control b4 or g4. There are a few other reasons: to create a luft when it’s not dangerous, or as a square for a knight or bishop.