Translation:What will you do the year after next?
I think you very clear! This is how I would write it in English:
If 'this year' is 2017, 'next year' is 2018, and 'the year after next' is 2019.
'The year after next' does not include 2019 and 2020, 2021... These would be the 'years after next' or 'every year after next'.
"How should it write" should be "How should I write it"
I hope that was what you were looking for!
The audio for さ来年 is difficult to understand. Compare with the recording at https://jisho.org/search/%E3%81%95%E6%9D%A5%E5%B9%B4
Or old words! I am having a seriously difficult time at those "hear and type" questions. [Although I am glad they have been added.] While I understand that not enunciating the individual words is indeed "normal spoken Japanese," I wish for us learners that they would clean it up (especially when we click "slow" speak). [Frequently, the only way I can match the choices to the sentence is to select some of the choices and just listen to how the computer pronounces those.]
I think the prevalence of "will" over "be going to" has a lot to do with how the English course for Japanese speakers was structured and how Japanese students are taught that "will" and "be going to" have different meanings (even though they don't always). Definitely worth an error report.
Here in the southern US, it's standard language. Having said that, I do hear disagreement over what it means. To me the "after next" denotes a two unit increase. That is, the year after next would be the year after the next year (current year plus two). But some people in other parts of the country use the term to simply mean next year.