The original fuzz bass effect, in "Don't Worry" by Robbins, for example, is one of these. Another example would be a click track being audible within the song; this I have also heard. "Lo-fi music" is music that lives on allowing such mistakes to persist, but more granularly, what do you call a mistake that is so allowed?
An excerpt from the wikipedia page on the Traveling Wilburys, a rock "supergroup" of sorts:
The term "Wilbury" also originated during the Cloud Nine sessions. Referring to recording errors created by faulty equipment, Harrison jokingly remarked to Lynne, "We'll bury 'em in the mix." Thereafter, they used the term for any small error in performance. Harrison first suggested "the Trembling Wilburys" as the group's name; at Lynne's suggestion, they amended it to "Traveling Wilburys"
Hardly well accepted terminology, but a word for it nonetheless.
Not limited specifically to music or recording, a happy accident might be called serendipity.
Several things that end up as effects in recording started out as epiphenomena, e.g., a lot of what we have come to expect in guitar tone came about because the process of sensing and amplifying the sound was imperfect.
But those are not "mistakes" exactly.
The Wikipedia article on "Don't Worry" says that guitarist Grady Martin didn't like the sound (so at least he considered it a mistake), but the producer liked the effect and kept it.
In your example of the audible click track, do you know that the artist or producer perceived it as a mistake? It could have been done intentionally (or at least deliberately).
Some "lo fi" might arise from accepting mistakes, but some might just arise from experimentation by someone with an artistic ear and a willingness to eschew conventional recording or performance wisdom.
I've not yet come across a good concise musical term for a note played accidentally during improvisation that ends up working out. We need one.
A related question was posed on the English Language & Usage stack: What is it called when something you previously took to be a mistake turned out to be the correct decision?
You might also look into what some call "Persian flaws", deliberate mistakes in handmade Persian rugs and carpets. (Similar concepts occur elsewhere in the arts.)