Other examples may clarify why it does not work: I do running? I do biking? It DOES communicate the same thing. But that does not mean it is correct english. I bike. I run. I read. I will do some biking. I can do some running. I should do some reading. All these are more correct English. (not necessarily better at at communicating.)
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Yes, you may correctly say "do some reading". This form is different from "do reading", which is not correct.
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You can use 'do reading'. for example:
"I need to do some reading on the subject" or "The homework for tonight is to do some reading'