To the extent he's not just playing against engines, I suspect he's looking at better models than this. He never plays Old Stodge (Nc3) like Chigorin does here.

As he did so often, Pillsbury got himself into trouble by playing overaggressively to gain the initiative. Not many Magnus opponents will do him that favor. Even so, if Pillsbury had interpolated 25....Qe1+ he could have gotten a tenable ending: 26.Kh2 Rxa7 27.Qb8+ Kf7 28.Qxa7+ Kg6 29.Kg3 h5 30.Qb8 h4+ 31.Kxh4 Qe4+! 32.Kg3 Qxc2 and White is going to have a hell of a time getting anywhere.

39.Qe8 could have transformed a win into a loss. 39....Qg6+ 40.Qxg6 Kxg6 and the pawn ending is hopeless for White even though he is a pawn up! The black king either penetrates at e3 or goes to g3, followed by the advance of the h-pawn and its exchange for the g-pawn. Black then wins the f-pawn and marches his g-pawn up the board. White can't do anything with his q-side majority.

But thats the norm around here

I let my Houdini program review 39....Qg6+! to a depth of 34 ply, and it showed Black to be winning in all lines: 41.Ke2 Kf5 42.Ke3 Kg5 43.c3 g6 44.Ke2 Kf4; or 41.Ke3 Kg5 42.Ke2 Kf4 43.Kf2 g6; are just two of many similar continuations, all winning for Black.

Houdini indicates that at move 39, White did not have any winning moves. Here are Houdini's top two choices for White at move 39: (.15) (31 ply) 39.c3 Qg6+ 40.Ke3 Qg5+ 41.Kf2 Qg3+ 42.Kf1 Qf4 43.Qe1 Kg8; and (.12) (31 ply) 39.Ke3 Qh6+ 40.Kf2 Qc1 41.c3 bxc3 42.Qd3+ Kg8 43.Qxc3 Qf4 44.Qe3 Qg3+.

At move 25, Houdini indicates Black's two top continuations are: 25....Qe1+ 26.Kh2 Rxa7 27.Qb8+ Kf7 28.Qxa7+ Kg6; (.39) (30 ply) 29.Qd7 Qxf2 30.Qxc6 Qf6 31.Qxf6+ Kxf6; and 25....Rxa7 26.Qb8+ Kf7 27.Qxa7+ Kg6; (.46) (29 ply) 28.Kf1 h5 29.Qc7 Kh7 30.Qf4 Qg6 31.Qd2 h4. Both of these lines appear to give Black good drawing chances.

After 17 rounds, the race for the top places was extremely close:

Pillsbury - 11 3/4 - (Rd 17 - Pillsbury beat Tarrasch)

Maroczy - 11 1/4 - (Rd 17 - Gunsberg beat Maroczy)

Janowski - 11 - (Rd 17 - Janowski beat Scheve)

Teichmann - 10 1/4 - (Rd 17 - Teichmann beat Chigorin)

In round 18, Maroczy took the lead:

Maroczy - 12 1/4 (Albin lost to Maroczy)

Pillsbury - 12 (Pillsbury drew his 1st game with Chigorin, but lost the 2nd game)

Janowski - 12 (Marco lost to Janowski)

Teichmann - 11 1/4 (Teichmann beat Mieses)

In round 19, Maroczy increased his lead:

Maroczy - 13 - (Maroczy drew his 1st game with Mieses and won the 2nd game)

Pillsbury - 12 1/4 - (Pillsbury drew his 1st game with Marco and lost the 2nd game)

Teichmann - 12 1/4 - (Reggio lost to Teichmann)

Janowski - 12 - (Mason beat Janowski)

In round 20, Maroczy maintained his lead.

Maroczy - 14 - (Popiel lost to Maroczy)

Pillsbury - 13 1/4 - (Mortimer lost to Pillsbury)

Teichmann - 13 1/4 - (Teichmann beat Tarrasch)

Janowski - 13 - (Janowski beat Marshall)

In round 21, Maroczy closed out the tournament, by playing two draws with Tarrasch.

Maroczy - 14 1/2 - (Maroczy and Tarrasch drew both games)

Pillsbury - 14 1/4 - (Pillsbury beat Marshall)

Janowski - 14 - (Teichmann lost to Janowski)

There is a link to click for computer annotations of the game <Notes by Stockfish 8 v270317 (minimum 60s/ply)>, and 41....h3 is indeed an error, one of several by both men around this point. I think the idea of 41.f4 (which Stockfish also doesn't like) is Qg5, either exchanging queens or winning the h-pawn. Pillsbury presumably didn't see a good response and decided to pitch the pawn in the hopes of counterplay after ...Qd7. Unfortunately for him, Chigorin found the strong 43.f5 in response, after which the ending was quite hopeless for Black.

Going back to my earlier kibitz, 39.Qe8 did not turn a win into a loss, because as <P&2> pointed out, there is no win for White there. So it looks like Pillsbury's 25....Rxa7 was an adequate defense; it wasn't necessary to throw in 25....Qe1+ as I thought.