“I could really use a cup of coffee.”
You're sleepy because you only got a few hours of sleep the night before. You're at work and really want a cup of coffee. You say this because there's no coffee available in your office.
I could really use a cup of coffee.
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You say "I could use ___" to indicate that you really want or need something, like a drink, but for some reason you haven't been able to get it yet. In the example at top, you say "I could really use a cup of coffee" because you're very sleepy and need some coffee, but there's no coffee nearby.
Here's another example - when you're struggling with something, and there's someone nearby who isn't helping you, you can say:
I could use a little help.
A similar phrase is "I could go for ___." Only use this to talk about things that you suddenly want, like food or drinks:
I could go for a beer right now. You want one?
"Coffee" is usually an uncountable noun, so you count amounts of coffee using the cup:
I drink 2 or 3 cups of coffee in the morning.
You may sometimes hear people ask for "a coffee", but this is usually used when ordering coffee at a restaurant or cafe. In other situations, you should say "a cup of coffee".
Also, "very" can only be used to modify an adjective:
It's very small.
I'm very tired.
"Really" can be used for adjectives as well as verbs:
It's really small.
I'm really tired.
I really wanted to go
Do you really think so?