The phrase, 'went out', when used along with the words, 'lights' and 'lamp' means 'stopped working or stopped shining'. As this phrase conveys a clear meaning when used in the given sentence, option C is correct.
The phrase, 'went away' means 'moved or travelled away from a person or place'. Eg: My wife went away to London without informing me.
The phrase, 'went off' means ' exploded'. Eg: Within seconds of lighting the spark, the bomb exploded.
The phrase, 'went over' means 'checked something carefully'. Eg: She went over the report twice, but couldn't find any mistake.
Given the meanings as above, the phrases given in options A, B and D cannot be used in the blank to form a meaningful sentence. Hence, these options are incorrect.