Taking a tougher stance on the prosecution of gun possession and robbery, the Manhattan district attorney on Friday officially revised several policies that had been fiercely criticized as too lenient, marring his first month in office. Days after taking office on Jan. 1, the district attorney, Alvin Bragg, instructed prosecutors in a memo to avoid seeking jail time for all but the most serious crimes. The document prompted weeks of pushback from police officers, small business owners and public officials, making Mr. Bragg a political target.