This answerThis answer explains how the HNQ algorithm works. The basic math is as follows:
$$ \frac{(min(\text{AnswerCount}, 10) \times \text{QuestionScore}) \div 5 + sum(\text{AnswerScores})}{max(\text{QuestionAgeInHours} + 1, 6) ^ {1.4}} $$
This means the two main things that drive the HNQ score up are QuestionScore and AnswerCount.
On Puzzling, a multitude of answers generally indicates a poor puzzle. Either it is not well-defined, and thus open to interpretation, or it is too ambiguous, and has many potential answers that could fit.
Here are some numbers to show how this works (these all assume a question age of < 6 hours):
$$\begin{array}{c|c|c|c} \text{QScore} & \text{ACount} & \text{AScores} & \text{HNQ score}\\\hline 10 & 3 & 12 & 1.465\\ 10 & 4 & 16 & 1.953\\ 20 & 5 & 20 & 3.256\\ 40 & 5 & 20 & 4.884\\ \end{array} $$
In general, the Question Score is much higher than the Answer Count, so I would venture to say that it is the single-most-influential data point in the calculation above.
So to prevent "the wrong type" of question from getting into HNQ, downvote1 them aggressively!
If you see questions that you think are of inferior quality, use your downvotes. That's what they're there for.
Some people may not be aware that downvoting a question costs you nothing. You lose 1 reputation for downvoting an answer, but question downvotes are free, so if you don't like what's appearing on HNQ, downvote1 it!
Oh, and did I mention that you can downvote1 inferior questions?
##Downvote - Downvote - Downvote - Downvote - Downvote - Downvote
1 NOTE: Especially with new/low-rep users, it can be very helpful to post a comment when you downvote, explaining why you think the question is not suitable for this site. (Thanks, @BmyGuest)