My child was left back in NYC for his behavior and I'm trying to find out if there is a test that he can. Take to get him into his right grade

No, there is no single test that would move him ahead a grade, Please understand, according to school, he is in the "correct" grade. Age to grade has little to do with it. If he was held back, it was most likely due to poor performance.

If he is now making straight A, having no trouble did great on end of school exams, you can approach the school to view him and see if he can be moved ahead.

Due to subject material learned, like pre algerbra or other math skills, often the higher up in school they go, moving up or missing a grade does not work well.

Tell us about how well the child does in school, what level they are in now.