- Read after his finishing homework.
- Read after he has finished homework.
Are these sentences correct? What's the difference between them?
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Neither sentence is correct.
Each is missing a subject for "read."
If the subject is "he" and the verb is describing repetitive action in the present, then the verb must be inflected to "reads."
If the verb is describing future action, the verb should be "will read."
Finally, in the first sentence, the possessive pronoun "his" must immediately precede "homework," the noun possessed.
"He reads (will read) after finishing his womework" is correct as is "He reads (will read) after has finished his homework."
In the first sentence, the word, "finishing," is converted from a verb into an adjective, so it describes what type of homework he is doing, instead of what he does to his homework. Sentence two is more correct, but still not perfect. You don't have a descriptor on "homework," so it's kind of vague and generalized as to what homework is being done. He may just need to finish some homework, but not all of it, he may be finishing somebody else's homework (though context suggests that's not true, nothing in the statement refutes it).
The best way to say it would probably be, "Read after he's finished his homework." (Typically one will assume that this means that all of today's homework is done, but it is only implied.)
This isn't a complete sentence either. We don't have a subject in it. Who is reading, and when? Is it that he is permitted to read after he did his homework, or that he did read after he completed his homework? These would be:
He can read after he finishes his homework.
He read after he finished his homework.