And Mr. de Blasio’s comments come as he is seeking a louder voice in national Democratic politics. The mayor is flying to the early nominating state of Iowa this week, part of a two-day Midwest tour designed to raise awareness about income inequality, his signature issue.

Representatives for Mr. de Blasio and Mrs. Clinton declined to comment on Sunday. Privately, the mayor’s advisers have rejected the notion that Mr. de Blasio’s national effort is about raising his profile, describing it instead as a grass-roots movement to promote the policy ideas, like higher wages and affordable housing, that he believes can narrow the nation’s wealth gap.

Inside City Hall in October, some three dozen commissioners and administration officials gathered to discuss children’s issues, rattling off statistics on cognitive development in toddlers.

The meeting, according to several people there, continued after a round of applause. But administration aides, taken with Ms. McCray’s impromptu performance, resolved to grow her audience: A recording of her lullaby will soon be featured in a promotion about early childhood development.

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.”