Donald Trump warned Thursday that his willingness to help hurricane-battered Puerto Rico was not unlimited, prompting a furious backlash, with the mayor of San Juan branding the president a "Hater in Chief."

Puerto Rico is struggling to recover after Hurricane Maria ripped through the island, leaving 44 people dead and cutting power and running water to much of its population, and its governor this week appealed to Trump for billions in additional federal aid.

"We cannot keep FEMA, the Military & the First Responders, who have been amazing (under the most difficult circumstances) in P.R. forever!," Trump tweeted, in one of a series of morning posts expressing frustration with the situation in the US territory of 3.4 million people.

Trump has pushed back hard at criticism of his administration's initial response to the disaster, accusing the media of exaggerating the devastation and minimizing relief efforts led by the Federal Emergency Management Agency (FEMA) with support from the Defense Department.

Earlier Thursday, the president quoted a journalist with the conservative-leaning Sinclair Broadcast Group as suggesting the island shared in the responsibility for the plight of its citizens.

"'Puerto Rico survived the Hurricanes, now a financial crisis looms largely of their own making,' says Sharyl Attkisson," Trump posted. "A total lack of accountability say the Governor. Electric and all infrastructure was disaster before hurricanes. Congress to decide how much to spend," he added.

- House approves aid package -

As Trump fired his latest rhetorical broadside over Puerto Rico, the House of Representatives approved a $36.5 billion package for disaster-hit areas including the US territory.

The measure outlining "supplemental" disaster spending passed 353 to 69, with all votes in opposition coming from Trump's Republican Party. The bill now heads to the Senate.

Maria slammed into Puerto Rico as the island was already struggling with a severe financial crisis, which forced the government to file for bankruptcy in May.

Carmen Yulin Cruz, mayor of the island's capital San Juan who has publicly feuded with Trump in recent weeks, pushed back hard at the president's latest remarks, charging that he was incapable of "fulfilling the moral imperative to help" Puerto Ricans.

"@POTUS your comments about Puerto Rico are unbecoming of a Commander in Chief they seem more to come from a 'Hater in Chief'," she tweeted.

House Democratic Leader Nancy Pelosi joined the criticism, tweeting: "It's truly sad to see @POTUS dismiss the suffering of Americans in #PuertoRico & #USVI. We must give more help, not less!"

- Aid still flowing in -

Last week, after visiting the island to view relief efforts, Trump had asked Congress to approve an emergency aid package of $29 billion for Puerto Rico.

With over 5,700 people still in shelters and much of the territory still without electricity and running water, Puerto Rico's Governor Ricardo Rossello this week asked Congress for more help.

Rossello said he has asked Trump for an additional $4.9 billion under the Community Disaster Loan program.

The governor's office did not respond to queries about the total amount of aid requested, which is now equivalent to around half of Puerto Rico's debt of some $73 billion.

Trump's warning prompted Rosello to tweet back: "The U.S. citizens in Puerto Rico are requesting the support that any of our fellow citizens would receive across our Nation."

Captain Scott Miller, a spokesman for the US military's Northern Command, which is overseeing Puerto Rico relief operations, said there had been no instructions to dial back aid.

"Our focus has continued to be in supporting FEMA and the Commonwealth of Puerto Rico in deployment of logistical support forces, commodities and medical capabilities, and we certainly still have air missions that are moving food and water and other vital capabilities into Puerto Rico," Miller told AFP.