Can I skip "this" here? If skip it how does the meaning change?
It would not make sense to omit "this" here because it is not grammatically correct. You could replace it with "the", but either "the" or "this" must go in front of "whole time". This is because "time" is a noun here so it must have an article in front of it. Since "time" is a specific noun, you can use "the" or "this" in front of it.
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