Under state law, the right to have an attorney present typically does not come into play until the arraignment hearing. Only at arraignment does the presence of legal counsel at each critical stage of a criminal proceeding become necessary. During the arrest process in New York City, the police can hold a person without the presence of their attorney for up to 24 hours. In some cases, this 24-hour window before arraignment is required can increase. During this time, the police can try to question or interview the suspect about the offenses they are alleged to have committed.
That said, a person that has been arrested has the right to an attorney if they explicitly request one or communicate to the police that they have hired. Once a defendant notifies the police that they are represented by counsel or asks for an attorney to be provided, law enforcement may not question them further without their lawyer present.
Even without an attorney, a person that has been arrested is under no obligation to discuss their case with the police. A person charged with a crime has the right to refuse to say anything outside of providing basic biographical information like their name and address. In fact, the police must notify a defendant of these rights when they get arrested. This is known as reading  Miranda rights.