The word "use" here is meant to convey the idea of "benefit from the use of." You could also say, "I could go for a nap." Same for, "I could use a drink." In the sentence, "I could use a nap," pronounce the word "yuz;" in the phrase, "benefit from the use of," it's pronounced "yuse."

'Use a nap' is an expression used to indicate that you are tired. You are suggesting that you're so tired that you could fall asleep. Alternatively it could mean that you're so tired and could use a rest. i.e. you just want to sit down and take it easy for a while. So it doesn't have to necessarily mean that you want to fall asleep. Here a few examples of when you could use this expression: "I'm so tired, I've been working non-stop I could really use a nap right now" "I've been on my feet all day, I could use a nap" Hope this helps. Let me know if you need any further clarification

in this case, the word "nap" is a noun. it means a brief period or moment of sleep anytime during the day - 睡午觉 If you need a nap, you can say "After waking up so early this morning, I could really use a nap!" If you want to take a nap, you can say "I am going to take a 15 minute nap."