The word はっきり (hakkiri) exists only in hiragana.This does not mean it is unimportant. Rather, it simply means that its roots are entirely native Japanese.
This word is most often pressed into service as a “suru” (する) verb, which means, adding “to do” (or conjugations thereof) at the end. So, we get this:
This means, to make clear, to make plain, to make distinct.
In other words, to make things plain, to make things clear.
Annoyed townsperson: “All I saw at the time was a red hat… ah!! はっきりした!! I know exactly who the thief is now!!”
This uses a past form (“hakkiri shita”) to say, outside of English grammatical order, that it is plain, and clear, what was true in the past.
Put another way:
Random townsperson: “All I saw at the time was a red hat…. ah!! I know exactly who the thief is! I’m sure of it!!”
Now, to hakkiri suru is to make something plain in the present and near future. Other verb conjugations can be used to further refine the use.
If you want to hakkiri sasetai, this is adding saseru + tai. This means, you want something made clear… by someone else.
This isn’t just a phrase used in detective mysteries (though it is). One of the most important themes in Japanese popular culture is sincerity, after all. People want to see others face up to their feelings (because that’s hard) and bring drama to a conclusion.
For example, by having the main heroine confess her love to the main hero. Or something.
Romantic plots are something that the audience wants made hakkiri by the end of a show. They are not always, and audiences are usually disappointed if that is the case.