It's definitely not a specific week, but in English "in the week" isn't specific with a present tense verb. But I think there is a difference between "in a" and "in the" in this context. "In a week" implies that a lot of work is done/achieved in the course of a week, taken as a whole. "In the week" implies to me that that a lot of the time each week is spent working.
Sorry, neither version is really everyday English. "I work a lot (in) this week" would have to be contrasting it with a different week or weeks. E.g. talking about a 2-week work cycle: "I work a lot (in) this week but not very much next." With and without "in" are equal. I think what you're looking for is either "I am working a lot this week" or "I have worked a lot this week" both of which are good English. Adding "in" to those is just wrong. The only justification I can think of for "I am working a lot in this week." is if "in" means "indoors", perhaps said by a farmer doing a lot of paperwork. In this case there would be a definite pause between "in" and "this".