Perhaps I missed this somewhere, but a general question about time and grammar. Are the days and months supposed to be capitalized? For example, you are supposed to write "Tuesday" instead of "tuesday" in english.
The correct translation into English should be "I work from September to April." the words "during" and "period", in this particular sentence, would be odd in English. It simply is not a proper English sentence.
It is a proper English sentence, just not a common one. The question is trying to get us to use the word "perioden" in the context of months. Omitting that from the English sentence would not be helpful.