There's an existing proposal to let moderators opt-out questions from HNQ: Allow mods or gold tag badge holders to prevent question from being on hot network questions list
I would like to add: there is a category of heated questions that attract debate, often rude debate, lots of comments, a suspicious mix of upvote and downvote, and occasionally a meta question asking how to discuss the question better, that should not be picked up at all.
This has definitely come up on Academia.SE (meta: Why are we challenging the premise rather than answering the question (question on potential sexist remarks)?) and also tends to come up on workplace.SE. Academia.SE has also opened a discussion on whether it benefits the community as all. Questions that are not only hot but very polemical - in the above case the presence of sexism was contentious - should not appear in the sidebar. It hurts communities when they do.
The hot questions needs to be less blunt. You should not put the spotlight on something that is only "hot" because it is "heated". And this can happen upstream in the algorithm rather than relegating it to moderation.
The algorithm should watch out for signals like downvotes in addition to upvotes or high comment volume to tell apart "hot" from "heated." Machine learning is not magic but I think isolating the signal here is a use case for offline ML analysis to write business logic against. Some communities have this problem worse than others; perhaps more aggressive HNQ-suppression logic should be deployed there, and softer logic elsewhere.
