As a part of Mayor Bill de Blasio’s relaxed policy on marijuana enforcement, set to take effect next month, the New York Police Department won’t arrest those caught smoking in public who have an open summons warrant and will instead take the offender to court to resolve the outstanding ticket, a police official said.

When the new policy directing officers to issue criminal summonses to those caught smoking in public as opposed to arresting offenders was announced in June, city officials laid out several exceptions. Among them...