Mike isn't at home at the moment.
Can I use "at this time" or "on this time" in this situation?
What is the difference between them?
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You can use "at this time" in the place of "at the moment". "on this time" is not a correct phrase, ever.
"at this time" means the same as "at the moment" and refers to the specific instant when the claim is being made with an exception. When you use "at the moment" at the beginning of a sentence, it could refer to any moment in the past depending on the sentence. Example: "At the moment of the crime, ..."
"on this time" is just not proper English; time is not a physical object.