Is this a grammatically and syntactically correct sentence?
When you have got to at home you have to get on with the work straight away.
I want to say when a person arrived at home he has to proceed with his work immediately.
Generally speaking, you get to a place or you arrive at a place.
So, both of the following would be acceptable:
✔ When you have got to the mall . . .
✔ When you have arrived at the mall . . .
But the combination of both prepositions isn't used:
✘ When you have got to at the mall . . .
✘ When you have arrived at to the mall . . .
The same is true when using home, but with the caveat that, unlike other nouns, a preposition is normally not used with it. (This is because the meaning of home is more nuanced than most other nouns—perhaps because of its additional adjectival sense.)
So, we would have the following:
✔ When we have got to home . . .
✔ When we have arrived at home . . .
Sometimes, a preposition is used, but it's not as common. It's also less common to see got to home than it is to see arrived at home. (I can't explain why this is the case.)
✔ When you have got home . . .
？ When you have got to home . . .
✘ When you have got to at home . . .
The first sentence is fine, the second is syntactically okay (but might sound strange to some people), and the third is ungrammatical (it's never used).
As a more general comment, changing the tenses used in the example sentence would result in a more natural expression:
When you get home you will have to get on with the work straight away.