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"writes on" is correct and usable in written English.
It usually means to continue writing about a particular topic or subject. For example: "John writes on history with much enthusiasm and knowledge."
Exact(60)
Lauren Sherman writes on style for Forbes.com.
(At least, David Gonzalez writes on Lens, in one borough).
He writes on Slate.
Adande writes on ESPN.com.
Jan Benzel writes on Rendezvous.
He writes on several old typewriters.
As Robert Lusetich writes on Foxsports.com, ouch.
He writes on.
He writes on a laptop.
A woman writes on a white board.
Paul Wheeler writes on local politics.
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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