As Leigh has said, 'could do with' is used to indicate a need, not a wish, so the New York example doesn't quite work. There are some situations when you might say 'I could do with a trip to New York'. For example, if it's necessary for your work to have a business trip there. Or even if you are tired and bored living in a small town, and you really think it would do you a lot of good to see the bright lights of the city! And yes, you can use 'could do with' followed by a verb in the gerund. These are two really useful verbs which native speakers use all the time, so they're good ones for you to learn and practise.

You have got it right.But could do with means more of a need than a wish. I would not use "Could do with a trip to NY" so rather "This summer we could do with a pool in the yard."

I've never heard "could do with." Could you be thinking of "could deal with?" That means that you wouldn't mind it, or you could handle it. For example: "If they would just turn the music down a little, I could deal with the heat." I think that "head to" is more commonly used. It means the same thing.

I've never heard "could do with." Could you be thinking of "could deal with?" That means that you wouldn't mind it, or you could handle it. For example: "If they would just turn the music down a little, I could deal with the heat." I think that "head to" is more commonly used. It means the same thing.