¿Qué significa Insist away ?

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It’s pretty difficult to explain the exact nuance, but I’ll try.

In this context, (verb) away = don’t hesitate to (verb). Another example:

I’m so sad.
Here, have a beer.
I can’t drink, I’m driving tonight.
Don’t worry, I’m sober tonight, I’ll drive you home. Drink away.

The “away” is there to show that the object of the action is “going away”. You drink the beer - the beer is being finished - the beer is going away - you’re drinking the beer away.

Naturally, this is used for any verbs imply that the amount of the object is decreasing, (e.g - drink, eat, rot)

However in modern conversational English, it’s often used with other verbs, just for the effect of “doing it with effort” - as if you’re (verb)ing so intensely that the thing you are doing it to is going to decrease.

Consequently, in this context, “insist away” is just a sarcastic way of saying “continue to insist”, hence “being irritated by that”.

This was an entire essay. Enjoy reading it.