In official correspondence games the computer assistance is allowed so most (if not all) of the players usually start their analysis with the computer suggestions (Stockfish, Lc0 or others). Some players limit themselves to this and play the engine's move, others try to improve with their own contribution. If no human contribution was possible, correspondence chess would become an hardware fight while results show that the best players can defeat "naive" opponents that rely on computer suggestions. In this sense, every correspondence chess win is a win over the opponent's hardware and engine.
Isn't it possible that you're not improving upon the engine's suggestions, but instead, your opponent is choosing suboptimal non-engine lines, and your engine is beating their weakened engine?
Occasionally it is possible. After seven years and more than one hundred games played I can tell you that I have been surprised by my opponent's reply not more than an handful of times. For "surprised" I mean he didn't play the top choice of the engine. In fact most of the times the best move in a given position is easily agreed on by any reasonable engine on any decent hardware. In few critical moments in the game, the best move is not clear and there are two or three or more playable alternatives that take into very different positions. In these cases the computer, after a long thought (one or more hours) usually converges to one suggestion and sticks to it even if given more time (a sort of "horizon effect"). These are the moments where a human, after a long thought, can overcome the computer suggestion and favor the 2nd or 3rd choice of the engine. So in brief no, I can't recall a game where I've been gifted the win by my opponent "weakened" move while most of the time I have confronted with the "engine's approved" suggestion and had to build my win by refuting it.
I assume that when you come across one of these novel moves, plug it into the computer, and give it time to search, it ultimately decides that it's superior?
Relatedly, can you give some examples of novel non-engine lines that turned out to be better than engine lines?
I'm interested because the experience in Go is humans simply can't keep up.
What is the evidence that it isn't a hardware or software differential between the players? I can't think of an easy way to ensure that both players started with computer-suggested moves of the same quality.
There are a lot of engines with rating on the chart way higher than the best humans, so every suggestion on their part should be in theory enough to overcome any human opponent. In practice most (if not all) of the players rely on Stockfish and Lc0 (both open source). During a game, most of the time the "best" move is easily agreed on by every reasonable engine on any decent hardware. Only in few cases during a game, the position offers two or three or more playable choices. In these cases a stronger hardware or a longer thought rarely makes the computer change his idea. It's a sort of horizon effect where more power doesn't translate into a really better analysis.
For example in a given position you could have 3 moves
M1 - a calm continuation with a good advantage
M2 - an exchange sacrifice (a rook for a bishop or a knight) for an attack
M3 - a massive exchange of pieces entering into a favorable endgame.
If the three choices are so different, the computer usually can't dwell enough to settle on a clear best move. Instead the human can evaluate the choices until one of them shows up as clearly best (for example the endgame can be forcefully won). In these cases the computer suggestion becomes almost irrelevant and only a naive player would make the choice on some minimal score difference (that can unpredictably vary on hardware, software version or duration of analysis). So the quality of the starting suggestion is somehow irrelevant if you plan to make a thoughtful choice.
Source: I’m a corrspondence chess international master