It actually sounds more natural when "that/when" is omitted from the first sentence "Every time the phone rings, he goes nervous." Also, it is correct to replace "goes" with "gets", so the correct and natural sentence would be "Every time the phone rings, he gets nervous." As for the second sentence, "He came at a time when we needed him most", it is more appropriate to keep the word "when" rather than using "that" or omitting it. In fact, the sentence would sound more natural to omit the phrase "at a time". This would make the most natural form of the sentence be "He came when we needed him most". Hope this helps! Let me know if you need an explanation.

Hi Min, In your first sentence, omitting "when/"that" sounds fine. However, we don't say "goes nervous", but rather, "gets nervous." Therefore, the sentence should be "Every time the phone rings, he gets nervous." As for your second sentence, omitting "when" sounds unnatural. As for replacing it with "that", that also sounds unnatural.