If you see something in the HNQ that you don't think should be there, you have several options, everything from editing to closing to flagging for moderator attention.
If you think the question is on topic but unfixable, please raise a flag. If you also happen to see one of the mods for that site in chat, feel free to let them know about it but please, also flag. Chat messages are not something that is easy to track after the fact to know why something was removed and we don't have a way to let moderators indicate why they removed a question at the moment. Using a custom moderator flag addresses all of this. A generic "remove from HNQ flag" would not do this.
I've been wanting to compile a list of solutions for various situations. There are a couple of things that we've already implemented to address problematic content in the HNQ and there's some things that are in my list of stuff to consider as future upgrades to the HNQ list.
Stuff we already have:
Per-site and network-wide title blacklists
If there are words showing up in titles that should be prevented from showing up in the HNQ list on either a per-site or network-wide basis, we can add those words to the blacklist to prevent the questions from ever showing up. This is a pretty blunt tool so we need to be sure that the questions will be considered pretty universally bad for the network. There are likely many NSFW words we could add to this list. If there's something specific you think needs to be added to this list, please make a Feature Request about it on the relevant meta site.
Question editing - particularly the title
If the question is a good fit for the site but the title itself contains problematic content, editing the title and the body to clean up that content is a good start.
Users on the site can close a question to get it off the HNQ list. Now, this requires that the question be a bad fit for the site for more than just the content - too broad, unclear, or off topic - but it's definitely an option that should be considered.
Moderator question removal
Flagging for moderator question removal is the best way of getting attention to a problematic question to have it reviewed for exclusion. Do remember, though, that moderators are not always on the site, particularly on weekends and especially on holidays. The options above this are preventative or something most users on the site can do to address the issue. They're preferred both because they don't require moderator intervention and they can be undone. Even a question flagged for moderator attention may only be edited or closed if the moderator feels that's the correct solution.
Stuff we don't have that might be useful:
Delay before a question can be added to the HNQ list
Right now we're working on adding a delay before a question can be eligible for the HNQ list. This means that a question will need to be at least 6-8 hours old before it can appear on the list, even if it qualifies otherwise. This is designed to give the users on the site the opportunity to vote on, edit or close the question. If a question is problematic, you can vote it down as that will prevent it from showing up on the list and since it can't show up in the first few hours, the question (and answers) will get community votes rather than network-wide votes that tend towards upvotes.
Give moderators (and other users) a heads-up when a question appears on the HNQ list
There are many ways we could implement this. I personally favor a HNQ hub of some sort - similar to the bounties tab - that would only be visible if there were at least one HNQ list question on the site. This would give a quick overview of what questions might need some attention. In conjunction with the previous tool, we could include new questions that will be added to the HNQ list once they meet the minimum age requirement. This allows users to vet the questions more closely before they are added to the list.
I'd also want to make this list a bit more sticky than the regular HNQ list. The HNQ list updates every 15 minutes, which means questions at the bottom of the list may come and go several times. I'd probably want this to capture all questions that have appeared on the HNQ list within the last hour or two unless they've aged out completely or been removed by a moderator.
There's a suggestion on the announcement post (and in a comment on the question here) to make this light up the diamond indicator like the Meta posts do. I understand that but it's not my first choice for solution. I don't personally see a reason to limit this list to moderators only. Such a suggestion also has a risk of being noisy on sites with many HNQs, particularly if that site rarely has controversial ones.
Let mods indicate a reason for removing a question from the HNQ list
This would help us see connections better - why was a question removed, is there something about the question that we could have filtered for to prevent it from showing up? Without it, we're having to guess. Though, in the interim, moderators wishing to explain removing a question from the HNQ list can flag the question themselves, explain the reason, and then handle the flag.
There may be other stuff I'm missing (it's early and I'm half-awake) but these are things I think will help a lot. If you have ideas, feel free to add them to my announcement post of the HNQ update.
I'm hoping that the minimum age update will be out later this week or next, so that should be good to see what impact it has.