I'm writing a fictional book set in NYC. As a part of the plot, there's public addressals on the on-going case. I'm just wondering what laws/policies pertain to what can and cannot be said about the case. I assume there are some laws, given that certain pieces of knowledge about the investigation could be beneficial to the killer(s).
In regard to what a police official, investigator, prosecutor, or other public official might say about a crime or an investigation of a crime, this is generally not a matter covered by law. Police department policy would generally be opposed to disclosing information that might help a criminal avoid apprehension. So would the policy of the office of the prosecutor. An official who violated such policy might be subject to internal discipline from his or her superiors.
A sufficiently senior official would be able to decide when such policy did and did not apply, and make or authorize public statements that might seem unwise to others.
Prosecutors have, in the past, made public statements that police thought helped criminals, but usually there is no way for the police to do anything about such statements, except perhaps apply political pressure.
A public official making a public statement about a crime or criminal investigation should be careful avoid anything that might constitute defamation. It is possible, although unusual, to sue such an official for defamation and win.
A person writing a non-fiction book (in the US) about a crime or the investigation of a crime can include any public statements or information about the crime or the investigation. Such a person can also include any information that person has learned by private investigation, or by interviewing or talking with anyone who knows about the subject. All that is protected by the First Amendment.
Only if the author learns some fact from a person who gives it in return for a promise not to use it might there be an issue, and even then such a promise might well not be legally binding. Such a promise would be quite unusual. Far more common is a promise not to identify the source of a fact. Such promises, whether they are legally binding or not, are strictly respected by ethical journalists.
Generally, when police with-hold information from the public regarding a crime, the reason is to keep information that "only the killer would know" from becoming public knowledge so that when they say "Mr. Doe was killed while making his dinner." A suspect who claims to have no knowledge of the crime saying "I never liked the guy, but I would never poison him" might tip off detectives that this suspect knows more than he's talking about if poisoning was the method of death... they never said the killer used poison.
This can be tricky as they need to show that they never mentioned poison in any public statement or in conversation with anyone not involved in the investigation and the suspect brought this element up on his/her own without any prompting in the conversation.
For a real life case, during the 2002 D.C. Beltway Sniper investigation, the snipers left a Tarot Card and the words "Call Me God" which, despite the evidence getting leaked, the phrase was misquoted. Because of this, when the sniper did call the tip lines and used the phrase "Call Me God" the police were able to identify the code and listen in. Other communications happened in secret, with at least one cryptically being referenced in a police press release hours before the snipers were caught.
Typically, police tend to keep details of the investigation closer to the chest than prosecutors once the suspect is captured, as the prosecutor has a job of telling their side of the story at trial and having the podium and access to the information can help the prosecutor get his story out to a wider audience. The knock-on effect is that it may be difficult to find a jury that can be unbiased as the public may have heard the details prior to their service in major criminal cases.