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'some slack' is correct and can be used in written English.
You can use it to mean "to give somebody or something more time or freedom to do something," as in the following example: I'm not expecting any miracles from my team, so I'm willing to cut them some slack if they don't get the job done right away.
Exact(59)
Cut us some slack.
Cut him some slack.
You cut yourself some slack.
Let's cut the guy some slack.
Will nobody cut Rannie Bookman some slack?
Well, her neighbors cut her some slack.
So please, cut us some slack.
I have to cut FreedomPop some slack.
So cut Richard Mourdock some slack.
So parents: cut yourselves some slack.
They've got to give him some slack.
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Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.

Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com