The House overcame a conservative rebellion Thursday to pass a new round of disaster relief, saying the $36.5 billion is needed to replenish funds that are quickly depleting in the wake of hurricanes that have battered Texas, Florida, the U.S. Virgin Islands and Puerto Rico.

The overwhelming 353-69 vote came even as President Trump suggested federal emergency workers cannot remain in Puerto Rico “forever” — and Carmen Yulin Cruz, the mayor of San Juan, Puerto Rico, accused Mr. Trump of “genocide” for what she said was an antipathy toward her island territory.

“I ask every American that has love, and not hate in their hearts, to stand with Puerto Rico and let this President know we WILL NOT BE LEFT TO DIE,” the mayor said in a statement she sent to Rep. Luis V. Gutierrez, Illinois Democrat, and asked him to publicize on Capitol Hill.

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“I ask the United Nations, UNICEF and the world to stand with the people of Puerto Rico and stop the genocide that will result from the lack of appropriate action of a President that just does not get it because he has been incapable of looking in our eyes and seeing the pride that burns fiercely in our hearts and souls,” the mayor continued.

The mayor and the president have been feuding for weeks, with Mr. Trump saying the federal government is fully invested, and saying the hurdles to recovery are on the island itself.

Earlier on Thursday, Mr. Trump, in a tweet, warned that federal emergency help couldn’t go on indefinitely.

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“We cannot keep FEMA, the Military & the First Responders, who have been amazing (under the most difficult circumstances) in P.R. forever!” the president said.

That set off a round of hand-wringing among reporters and Trump opponents who wondered what spurred the comment, and called it a “threat” to the territory.

One reporter, in a press briefing with White House Chief of Staff John F. Kelly, demanded to know whether Mr. Trump accepted that Puerto Ricans are American citizens and deserve the same help as Texas and Florida have gotten. Mr. Kelly affirmed both statements.

The reporter then demanded to know why Mr. Trump had tweeted “that we can’t be in Puerto Rico forever.” Mr. Kelly corrected her, saying the president had specified that “the U.S. military and FEMA can’t be there forever.”

“The minute you go anywhere as a first responder — and this would apply, certainly, to the military — you are trying very hard, working very hard to work yourself out of a job,” said Mr. Kelly, a retired Marine Corps general.

While Puerto Rico is claiming the lion’s share of attention, and is likely to get most of the money from the new disaster bill, it’s just one of the crises the legislation is intended to address.

There’s $577 million included to fight wildfires, with the death toll rising from raging blazes in California.

And the bill includes $16 billion to bail out the National Flood Insurance Program, which is the government’s troubled backstop program to provide relief for those who cannot get regular flood insurance because they live in too risky areas.

Some of the 69 Republicans who opposed the bill singled out the bailout as their reason for opposing the package. They said the bailout forces responsible taxpayers to cover the costs of those who insist on living in risky areas.

Other Republicans objected because the money is tacked onto the deficit, forcing the costs onto future Americans, rather than paid for through spending cuts or tax increases now.

The bill includes $18.7 billion to replenish the main disaster relief accounts that keep Federal Emergency Management Agency officials in the field.

The bill still needs Senate approval.

Congressional leaders and the White House said it’s not bill they’ll be asking taxpayers to pay for the disasters.

“There will be others,” promised House Appropriations Committee Chairman Rodney Frelinghuysen, New Jersey Republican.