Let's break this question into two pieces.
The exercise instructs you to cross out redundant words in the sentence. Redundant means not needed or superfluous.
I'd argue that the entire beginning of the sentence is redundant because someone reading the sentence already knows that you're writing to tell them something. In American English at least, you can also omit the and between come and stay as well (ht: Robusto).
I’m writing to tell you that I can’t come and stay with you the next weekend.
Next Weekend vs. The Next Weekend
I don't think there's enough context to know whether the sentence should be "next weekend" or "the next weekend," since they mean different things. I suspect it's the former, but it's just a guess, and it's worth a short discussion about the difference.
English has a slight, unfortunate ambiguity in how the near future is described:
- "this weekend" = current weekend or first upcoming weekend
- "next weekend" = first or second upcoming weekend
- "the next weekend" = first weekend after some previously referenced date
A: Do you want to meet next weekend?
B: Do you mean this weekend or the following weekend?
A: The following weekend.
B: Sorry, I can't make it. How about the first weekend next month?
A: Can't do it, but I can meet the next weekend.
B: That works. See you then.
So, getting back to your question, it's possible that "the next weekend" makes sense, if they're talking about the weekend following some previously discussed date. It seems unlikely, though, and I tend to agree that it should be "next weekend."
The same rule applies to your other examples. The apology happens "the next day", i.e. the day after the argument. The doctor calls next week, i.e. the first upcoming week.