I have a question. If my example sentence is: 친구를 기다렸는데 안 왔어요. And I have to write it with ~아서/어서, would it be correct to say: 친구를 기다렸는데 안 왔어.?

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친구를 기다렸는데 안 와서요. could be an exmaple of -아서/-어서 in this case. As '-아서/어서' is used to show a time-order relationship, I would assume that there must be an omitted phrase from the full sentence, say, 친구를 기다렸는데 안 와서 (그냥 집에 갔어)요. It is totally find to write in this way, and a reader will get the omitted information out of the context.
Hope this answers your question!