Yes, the sentence is grammatically correct. It uses "because" and "when" appropriately. However, for improved clarity, you might consider restructuring the sentence slightly: "I answered the question wrong on the exam because my teacher didn't teach me that." This revision maintains the same meaning but organizes the elements in a slightly clearer way.

Because my teacher hadn’t taught me that, I answered the question incorrectly on the exam. Because my teacher hadn’t covered that, I got the answer wrong on the exam. Note that using ‘wrong’ as an adverb as you have is possible, but less polished.