The Stack Exchange network has a "hot network questions" list that appears in the sidebar on every page. An algorithm picks posts for this list based on newness and velocity of activity. In 2018 this was completely automated and sometimes made bad choices. The community had been asking for a remedy for years, and got deafening silence.
In October 2018 somebody on Twitter complained about a question on that list, and the company reacted disproportionately, causing a lot of pain and aggrevation. They also finally asked the community for input about "revisiting" hot network questions. This is the answer I wrote. It was well-received by the community, though the company addressed only a tiny part of it. (Moderators can now withdraw posts from their sites from the hot list manually. That's it.)
The HNQ implementation has at least three problems:
It optimizes for controversy, not quality. Sensationalist questions draw lots of rapid response, which feeds the HNQ algorithm, and then once it's on HNQ it gets even more rapid response, which keeps it there (and also distorts Q&A on that site).
It's completely automatic; sites have no way to intervene and remove a question short of closing it. But closure is frequently the wrong answer; the question is perfectly valid, but controversial.
Everybody sees it, all the time. If that list on the side of the page has even one or two "exciting" questions, we can pretty much guarantee that some readers will be upset. (There is a regex filter on titles, but it comes with all of the usual regex challenges.)
If we want to have something like HNQ -- and I've heard enough people on smaller sites saying they value the promotion to go with that for now -- then we need to find a way to let communities curate the content. We should also consider user controls for those who need them. (I get it; I don't want my coworkers to see that question about Klingon mating rituals either.) We should also adjust the algorithm so it doesn't so strongly prefer controversy.
Here are some ideas along those lines:
To aid curation: provide a way, from the question page, to see if this question is currently hot or likely to become so. If automatically displaying that would be an expensive operation, let's add a way to check. Perhaps checking the hotness score and comparing to the minimum score from the HNQ is good enough; we don't need to actually do the lookup.
Automatically exclude questions that the community has signalled have issues: protected questions and questions with two or three close votes. (Probably we should make that number adjustable.) More thoughts on this here.
Give moderators and high-rep users a menu item for "exclude from HNQ" so they can be proactive.
Suggested in comments: allow communities to review hot questions (also suggested on other answers here)
Factor downvotes into the hotness formula.
I don't have concrete suggestions here, but let's review the velocity part of the formula. Lots of answers isn't necessarily a good measure. Lots of quality answers might be, but that's hard to measure. But let's factor answer score and length in more and number less.
Allow users to (persistently) collapse the HNQ, and start in the collapsed state.
As noted in comments, we should also review the effects of visitors bearing association bonuses on hot questions (should they be able to vote? comment?). I've been focusing here on how questions get (or don't get) to HNQ visibility, and with all the voting that's already happened I don't want to expand the scope of this answer now. Let's use other answers for that part of the problem.