The comments on the question quote my statements from the original announcement of this tool and I think that, for the most part, that's still correct. The main point of guidance is:
We're putting the power in the hands of our moderators to remove questions that don't set a good example for their sites. I recommend each site have a meta discussion with guidance for moderators about when - if ever - a question should be removed.
This is the overarching rule and what I still stand by. When we created this tool, I had no way of knowing how often it would need to be used so I deferred to the most knowledgeable people - the engaged users on a site - to help inform the mods about when to remove questions.
For most sites, the rest of my guidance was (and still is) good -
In general, we recommend that you exclude questions that attract negative attention to your sites, that is, questions that are controversial, start large amounts of debate or arguments or even edit wars. Removing a question should not be a substitute for fixing it! Remember that it may take several hours for a moderator to respond to a flag so do what you can, first:
- If the title seems click-baity or doesn't adequately describe the question, edit it!
- If the body is full of spelling or grammatical errors, fix them!
- If the body contains unnecessary detail or salacious content, see if it can be removed without impacting the question.
- If the question is unclear or broad, vote to close it. In most cases it will be better to close a question and wait for it to be improved rather than asking for it to be removed.
This tool is a big gun and should be used sparingly. Don't reach for it if you think the question can be fixed.
When moderators kick a post off the HNQ list there will be a delay of up to fifteen minutes or so as the list is cached but the question will be removed the next time it runs. In addition, an event will be logged in the post timeline and edit history that indicates when it was removed and by whom. This will help us understand what sort of posts are unwelcome in the HNQ list on different sites.
At this point in time, mods can only use this tool on a post currently in the HNQ list - they can't use it preemptively to prevent a question from being added to it.
This tool should be used relatively rarely... on most sites even getting a question on the HNQ list is pretty rare in the first place - that's why we added rules to set an upper limit to the number of questions from a specific site - to share the limited spots more around the network.
On some sites, it's very common - sites that have healthy voting cultures or where questions regularly get multiple answers. On some sites, those featured questions are still interesting and engaging - the concern that the mods on Math, specifically, have, is that the questions that tend to be selected are not the sort of question that help the site look good to visitors and look like a place where experts want to participate.
And I respect that reasoning. Not all questions can be excellent, clever ones but that doesn't mean that banal ones that just happened to get a couple of answers and an upvote should be broadcast to the entire network. This wasn't something I was thinking about when I wrote my considerations (quoted above) and I think it's a decision that is in the interest of the site, if that's what they feel is best for them.
So, the original advice stands - mods (or site users) should use meta to figure out what policy best suits a specific site rather than relying on one network-wide policy for guidance.