1. He is going to leave for New York.

2. He is goin' to leave for New York.
3. He is gonna leave for New York.

4. I want to go back home.
5. I wanna go back home.
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Can we use all these expressions?

One and four are correct. The others are all slangy and not correct grammar, although some people might use "goin'" and "gonna" in casual, informal speech.