You need to meet with a lawyer about the citizenship application. There are several different ways a charge in NY could be resolved in that manner, ACD, conditional discharge, etc. They have different effect on immigration and what was said in court is critical. You cannot proceed with a citizenship application without a lawyer to confirm the details of the criminal case and assess the immigration impact.
Your husband should schedule a consultation with an immigration lawyer. There are important details that need to be scrutinized, including but not limited to the nature of the charge, the underlying factual allegations, and the manner in which the charge was disposed of. If he applies for citizenship without going through this process, he will not only be jeopardizing his ability to become a citizen, but even possibly to remain a Lawful Permanent Resident. Keep in mind that a Naturalization cannot be withdrawn without consent of DHS - DHS will see it through, ... to the bitter end.
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If the arrest happened in NY it sounds like your husband received an Adjournment in Contemplation of Dismissal. This is neither a probationary period or a dismissal of the charge. What happens is after a year the charge will be dismissed and sealed unless the Prosecution for some reason objects. Your husbands status restored to what it was before the arrest, as long as he has avoided getting in trouble since the arrest.
New Jersey offers a similar resolution called a Conditional Discharge, however, they operate differently procedurally and should not be lumped together as "the same." I hope this helps and best of luck with your situation.