I have an exercise requesting me to cross out a word in each sentence if it is redundant. There is a sentence in that exercise:
I’m writing to tell you that I can’t come and stay with you the next weekend.
Is ‘the’ necessary here? Can I cross out it? I ask this question because sometimes people use the right before next:
We had a big argument, but he called me the next day (= the day after) to apologize. Cambridge dictionary
However, there are still some cases they don’t use the:
The doctor will call next week to check on your progress.
Best Answer
Let's break this question into two pieces.
What's Redundant
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).
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:
Example conversation:
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.