application for leave on friday
In Canada, I would simply write

Hi Tom,

I would like to take this coming Friday off.


But I think that in some countries you need to be more formal.
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Yes. I would say that to my supervisor but I would not put that in writing because it is very informal. Even in an email you should be more professional because it will be on record.
As I said, it depends on the culture of your country, and on the culture of your company.