English Translation
Now listen to me closely.
The use of “mal” is for me quintessentially German. No English word can ‘intend’ what “mal” does so simply in so many different German sentences.
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Yes 
“mal” is used so often, yet I can’t really explain how or when to use it. To foreigners, it must appear to be used randomly or arbitrarily 
There can be quite a difference between using and not using “mal”.
In this sentence, using “mal” gives you the hint that the speaker is angry.
Not using “mal” can have a neutral meaning. For example, the piano teacher points out to the pupil that now comes the part of the song where the pupil needs to listen to the teacher especially carefully. “Jetzt hör mir gut zu!”
If you use “mal”, which can in this situation be translated with “for once”, indicates some anger or frustration. For example, two persons (Alice and Bob) are arguing and Alice has the feeling that Bob never listens to her, and now Alice has had enough, and commands Bob to finally listen to her, for a change. “Jetzt hör mir mal gut zu!”
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I’ve been trying to work through how ‘mal’ indicates time in some way, and your pointing out how "mal’ can mean ‘for once!’ really helped. Thank you so much!
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Perhaps it helps to think of “mal” as “at least (this) one time”. (As in my example of the arguing couple I gave in my other post. “Now listen to me, and listen to me well! At least this one time.”)
But sometimes, you’d also have to add “sometime/anytime” because in German, it’s often left unspoken. For example:
“Wir sollten mal Kaffee trinken!” is short for “Wir sollten (irgendwann ein)mal Kaffee trinken!”
In English: “We should (at least one time) have coffee someday”.
It’s interesting how it seems that English and German leave different parts of the sentence unspoken: “one time” or “someday”.
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