Just one thought that might help. It is easier to draw a game if the other person has a R pawn such as the h pawn. If your opponent has any other P than a RP and they have opposition, they can win the game. K and P against K endings.
Should the pawn capture towards the center or not?

thanks for the thoughts so far. Yeah, I know King and pawn endings relatively well (studied them before, but like everything else in chess, it seems that even the simple can sometimes become complicated ). It makes sense that the computer is playing pawn structure for an endgame, but I suppose I just wasn't thinking endgame when the game barely began

I just don't like the backward e-pawn. Therefore, capture with the h-pawn. The open f-file for the rook is not enough compensation for the defective pawn structure.

I just don't like the backward e-pawn. Therefore, capture with the h-pawn. The open f-file for the rook is not enough compensation for the defective pawn structure.
Yes thank you SeniorPatzer, it looks like I gave myself a backwards pawn without considering its impacts enough. It is amazing how much of a role pawn structure can play in an entire game isn't it? Of course, I knew these ideas before, but in a game it is sometimes difficult to think of everything
p.s. although in hindsight, pawn structure is always something that should be considered. If any observer to this forum takes anything from this to assist their own chess game, then it should be the importance of pawn structure.

Ideally you want your pawns to stay together. After hxg6 you have one pawn island but after fxg6 you have 2. I also don't like fxg6 because e6 and e5 become weaknesses. You need your f-pawn to defend these squares. It's also possible that you could build up in the center at some point with f6 followed by e5. You can't do that if you play fxg6.

One question here, if it became a pure pawn ending does either capture win?

One question here, if it became a pure pawn ending does either capture win?
I don't think its a "winning K+P ending after either capture, if you're talking about an average everyday player like myself. But if you're taking Carlsen? Yea...that might make it "winning" But i do think it makes a difference with pieces on the board.

One question here, if it became a pure pawn ending does either capture win?
I dont think its a "winning K+P ending after either capture. But i do think it makes a difference with pieces on the board.
Yeah, I guess it does. I would also choose hxg6. The move fxg6 seems to weaken the structure too much.

Opening the f file would be good if it were coordinating with something.
It's like someone calling a venue, and paying a fee to have their band play a concert... but they don't have a band yet. It makes no sense. First get a band together, then start looking for venues.
Sorry for the analogy, I'm a little out of it at the moment.
I any case, higher number of pawn islands can make both your mid game and endgame weaker. It's a good sign you're willing to think outside of the box, but until you're a little more familiar with the underlying logic, it's better to stick to principals.

Ideally you want your pawns to stay together. After hxg6 you have one pawn island but after fxg6 you have 2. I also don't like fxg6 because e6 and e5 become weaknesses. You need your f-pawn to defend these squares. It's also possible that you could build up in the center at some point with f6 followed by e5. You can't do that if you play fxg6.
Very cool, put so succinctly. Thanks very much for that summary.
AND WOULD ALL LLAMAS TRY TO BEHAVE
freakily xD

Tactically, fxg6 is worth considering if you can actually make use of the f-file to pressure white. For example, in some positions where black is better developed, fxg6 followed by an immediate e6-e5 opens up the center and gives black the initiative.
In the position above, fxg6 makes no sense. Black can play for nothing on the f-file, nor does he have e6-e5 as a follow-up. Further, the e6-pawn is seriously weakened, which makes white’s e3-e4 push easier after 2. Bb3. The “now I’ve got a semi-open file with a rook on it and let’s see what’ll happen” mindset will never work above a certain level, where your opponent knows what little a rook can do in the middlegame.
You said that you “don’t follow exactly engine advice down to the centipawn”. It’s true that the machine has its flaws and is not 100% trustworthy. However, following line(s) suggested by the engine (which means playing these moves out on the digital board, see how the position and evaluation changes and figure out what the engine’s plans for both sides are) usually helps to understand the position.

Black has no realistic prospect of attack along the f-file, since it's too easy for White to defend f2 without making compromises. So after fxg6 Black has no real compensation for the obvious drawbacks (1) backward e-Pawn, (2) one fewer central Pawn, (3) two "islands" of Pawns instead of one, and (4) devalued Pawn structure.
After capturing hxg6 instead, Black avoids the drawbacks (1) to (3), and retains the possibility of pressuring White's King via g6-g5, g7-g6, Kg7, Rh8 and possibly g5-g4. The attack along the h-file is much more difficult to parry without making concessions (such as retreating a Knight to f1).

I don't think you can say anything about the reason why the computer prefers a move without actually looking at the lines it gives. I really don't understand why people don't just do that. The moves which the computer gives are more important than the evaluation. The evaluation is wrong in many cases.
The computer likes to play f4 as white and play moves like Ne2-c3 and Bc2 to put pieces on better squares. fxg6 gives black a weak e6 pawn and an open f-file. However if white goes f4 then there is no way to make good use of the f-file. On the other hand there is no way for white to really put pressure on the e6-pawn. The position is about equal.
hxg6 allows black to play f5 to reinforce e4 in case white plays f4. I think black stands better in the resulting position.

I won't go over old ground, but wanted to add a new, minor point, which hasn't been brought up yet. Generally, pawns have two squares which they can attack. They kind of create minefield squares for the opponent, and if the opponent drops a piece on those squares, the pawn can eat that piece. A rook pawn however has only one minefield square since it can only attack one square on the chessboard, and that square is on the edge, generally away from much of the action.

Personally, I would also prefer hxg6. The alternative creates more pawn islands, and even if the rook were freed it gets no significant play. Also, white has castled, so there are no immediate rook threats to the h-file, making it safe for black to take with the h-pawn and support the center.

Thanks you for everyone's insights; I think I have a much better thematic understanding for what I should be looking for in these situations than I did before I created this forum. This particular position I selected has apparently several subtleties favoring hxg6 it seems. I suppose I was more curious to the thematic questions I should ask myself in similar positions (which this forum has helped a lot with). I felt that similar positions often come up and I wasn't really understanding what was at play enough of the time compared to how I felt about other positions.

Opening the f file would be good if it were coordinating with something.
It's like someone calling a venue, and paying a fee to have their band play a concert... but they don't have a band yet. It makes no sense. First get a band together, then start looking for venues.
Sorry for the analogy, I'm a little out of it at the moment.
I any case, higher number of pawn islands can make both your mid game and endgame weaker. It's a good sign you're willing to think outside of the box, but until you're a little more familiar with the underlying logic, it's better to stick to principals.
Also, don't feel sorry for the analogy; I actually found it enlightening. This analogy resonates for me because I have been attempting to think "more outside the box" lately. Personally, I love the endgame and so I used to be constantly thinking "endgame." What I've recently come to realize is that sometimes you are giving up a dynamic edge to always play for such endgames. The element about the venues sparked my insight because I think this is the kind of thing I was looking for through this forum. If I can coordinate play then opening the file might be stronger, but if I can't do enough with it, then I am hurting my endgame chances.
I like how this is described, it sounds like a trade-off of attacking opportunities for now, versus pawn structure for a later endgame; it is sort of like an investment, but ultimately the chess player is trying to decide if it is a good deal or not (based on the position and if their play has compensation enough for the weaknesses)
In this position, I was Black. I know that the general rule is that pawns usually are best "capturing towards the center." In the position, I captured with the f-pawn since I wanted to open up the file for my Rook if I could. However, the post game analysis calls this an inaccuracy and gives the h-pawn capture about 50 centipawns better.
I don't follow exact engine advice down to the centipawn, but I am curious what is so appealing for the computer to prefer opening the h-file instead. I would agree with the h-pawn capture if not for the location of my Rook, but I thought this reason was enough to play f-captures. Is it simply the fact that side pawns are not as valuable (and the h-capture basically solves this problem), or is there thematically more at play here that I am not understanding?
Any thoughts on the position would be appreciated