Is it a draw?

Books usually says that this position is a draw but chess.com app (Android) says that white have won one centipawn; why?

+0.01 is not a centipawn but a centipoint. There is no relationship between the point scores of position evaluations (here +0.01) and the value points assigned to chess units to compare their relative values (like 1 for a pawn and 3 for a bishop).

Generally one needs an evaluation score >2.0 to be reasonably confident of a win, but every number other than +Mx, -Mx or 0.00 indicates the engine is not absolutely certain. 0.00 has dual interpretation, it either sees the position is a draw or judges that the position offers black and white equal chances.

The +0.01 score therefore tells you that the engine does not see the position is a draw but it does see that white's winning chances are negligible. Obviously not seeing the draw is the result of poor programming as the engine does see all the moves which make the draw inevitable. Different platform versions of the engine or a different engine might yield the proper score 0.00 - like StockFish on chess.com.

@3

"+0.01 is not a centipawn but a centipoint" ++ That is the same.

"There is no relationship between the point scores of position evaluations (here +0.01) and the value points assigned to chess units to compare their relative values (like 1 for a pawn and 3 for a bishop)." ++ There is: the evaluation is a modified count of material in units of 1 pawn.

"Generally one needs an evaluation score >2.0 to be reasonably confident of a win"
++ No. Per GM Larry Kaufman: > 0.7 white wins, < -0.7 black wins, between -0.7 and 0.7 a draw.

"0.00 has dual interpretation, it either sees the position is a draw or judges that the position offers black and white equal chances." ++ No 0.00 is a draw.
"Statistics of the previous two superfinals show that a Leela book exit of +0.30 or lower is an almost 100% certain draw. " - GM Sadler

If ever the 1.0 evaluation score was chosen to calibrate as the value of a pawn, it was only in the game starting position. Quite obviously, the relative value of a pawn is greater when there is less material on the board.

There is a big difference between dividing the scale in 3 colour shaded territories and "being confident of a win". Having followed lots of internet games I concluded many GM-level players accept higher scores than +0.7 for their opponents in their prepared variations which means either that the engines evaluations are poor or these scores are not as bad as they look. I guess the difference needs to be greater for weak players than for "perfect engines" and that is part of the interpretation. This also applies to Sadler's statement "+030 is an almost certain draw". For whom? Grandmasters I suppose. Or for engines? Or only for perfect engines? But he is not completely dumb as he says "almost". That is the restriction for positions with "endgame study" characteristics, see below..

You are most certainly wrong on "0.00 is a draw" unless engines have adopted it as a code, making sure that this value is only used for "calculated draws" and not "projected draws". To state that they are the same is the same as stating that engines are perfect. For instance, there are many endgame studies where the engine fails to see a way to unbalance a situation and therefore projects a 0.00 score. And then that turns out to be wrong. Which is very different from seeing a stalemate coming - or a repeat - or a fortress style stand-off. Btw, to claim a win in an endgame study variation requires an evaluation score of 2.5 to 3 to convince the judges. They won't buy into a 0.8 win verdict.

@5

"Sadler's statement "+030 is an almost certain draw". For whom?"
++ For engines. His comment relates to select imposed openings for TCEC to avoid all draws. They select openings above 0.30 and below 0.70.

When I look at Stockfish's analysis, it immediately shows 0.00 in all variations. It is hard to imagine where that 0.01 that the OP posted is coming from. I guess chess.com's "Game review" feature is the culprit (again).

It is a draw but only if black retreats directly backwards. If the black king moves back& L or R white wins. You could have played it one more move as white to see if black knows this. I suppose the computer says white is slightly better since white cannot lose and black can still blunder (assuming the evaluation is a probability based on some sort of ensemble of lines).

This is not an unreasonable thought, but engines don't think like that. As I said, Stockfish evaluates it as 0.00 instantly.

Thank you all for your very precise comments (that is still an understatement !).

Ah, an expert if I am not mistaken. As a retired computer programmer I recognize the scientific modelling involved. What surprises me is that SF would recalibrate its scoring system over time. Do all of its users know what that implies and adapt their preparation strategies? Or are the changes so minute to humans that it doesn't matter? Did other engines follow suit?

This new evaluation thing is for the new SF release, SF 15.1. I don't it has anything to do with the question at hand.

Not that much but it appears we already progressed from there. Just curious what they explained about a fundamental scoring change to the professional engine users like our chess title holders!