Many of Mr. de Blasio’s religious initiatives began as campaign promises, made during his courtship of evangelicals, Muslims, Orthodox Jews and other religious groups that now make up a substantial portion of the city. They are groups that often felt frustrated by the administration of Michael R. Bloomberg, a political independent who was avowedly secular and insisted on a strict church-state divide.

“I understand the role of civil libertarians,” the mayor said. But he added, “When it comes to the notion of public policy and what we are trying to achieve through public policy, I am going to be creative, in a very appropriate way, to get that done.”

Mayor Bill de Blasio was late again this morning. The Daily News reported that he arrived 15 minutes behind schedule to a St. Patrick’s Day Mass at St. Patrick’s Cathedral. This was after he somehow managed to be equally late to a St. Patrick’s Day breakfast and reception in his own house, Gracie Mansion.

“Happy” wasn’t the word earlier this month in the Rockaways, when Mr. de Blasio was about 20 minutes late to a St. Patrick’s Day Parade, skipped half the route and got booed by the crowd.

Mr. de Blasio, a Democrat, views ultra-Orthodox New Yorkers as a key political constituency, and he pledged to rescind the consent rule on the first day of his administration. His aides have spent months conferring with Orthodox leaders on a compromise.

The administration did not allow city health officials to speak on the record during a briefing about the new policy. Instead, reporters were asked to attribute quotations to “Official 1” and “Official 2.”

The fictions behind the mayor’s accounting include crediting the unions with savings achieved by purging insurance rolls of people who were ineligible for benefits — thereby reclaiming money that belonged to taxpayers, not to the workforce.

They also include inflating projected hikes in health insurance premiums, pressuring the city’s carriers and their state regulator to hold down increases and crediting the unions with the difference between the initial outsized projection and the final bill.

In recent days, Mr. de Blasio has sounded downcast, according to aides who have spoken with him. The manager of his mayoral campaign has returned to help. And his team is considering focus groups and a poll to refine the mayor’s message to New Yorkers who may have soured on him.

How Mr. de Blasio reached this point is a story of accumulated slights, political miscalculations and the slowness of members of his inner circle, almost none of whom have close ties to law enforcement, in reacting to the warning signs of what would become the gravest crisis of his young mayoralty.