(CNN) President Donald Trump's budget plans to cut the Children's Health Insurance Program by at least 20% over the next two fiscal years and slash Medicaid, which covers millions more children.

Millions of poor and working families could lose their health coverage if his proposed budget, released Tuesday and called "A New Foundation for American Greatness," gets through as-is. It would hit children's health care hard and break Trump's campaign promise to "save" Medicaid "without cuts."

I was the first & only potential GOP candidate to state there will be no cuts to Social Security, Medicare & Medicaid. Huckabee copied me.

Over the next decade, the proposed budget would cut Medicaid more than $800 billion, or nearly a quarter of what the health care program is projected to spend on poor families.

Poor children are covered by a complicated mix of programs. Medicaid covers 37 million children. The Children's Health Insurance Program, known as CHIP, has 8.9 million enrolled. Together, these two programs cover about one in three American children , according to the Kaiser Family Foundation.

Cuts to the social safety net

When Trump ran on a promise to end Obamacare and give people better health care, he promised not to touch Medicaid "like every other Republican," and yet Office of Management and Budget Director Mick Mulvaney said Monday that Trump personally approved each cut.

"I can tell you exactly how it went down," Mulvaney said. "I went into the President with a list of proposed entitlement reforms.

"He went down the list, yes, yes, no, no, yes, no, yes, no, no," Mulvaney said. "The nos were all Social Security and Medicare. That is it. He said, 'I promised people on the campaign trail I would not touch their retirement and I would not touch Medicare, and we don't do it.' "

The Republicans who want to cut SS & Medicaid are wrong. A robust economy will Make America Great Again! https://t.co/u25yI5T7E8 — Donald J. Trump (@realDonaldTrump) July 11, 2015

Mulvaney emphasized that the biggest savings in the budget come from cutting or scaling back entitlement programs.

The budget would also slash other programs that help the working poor. It cuts food stamps, student programs, the earned-income credit, the child tax credit and disability payments, among others, while increasing military spending, border security and infrastructure.

Asked whether these cuts are "hard-hearted," referencing cuts to an example he gave about an education program, Mulvaney responded , "You know, for the schoolteacher in Kenosha, Wisconsin, who is trying to raise two kids by herself, save for her college and save for their retirement, isn't it hard-hearted to go to her and say, 'give me money for a program that is only 6% effective'?"

Of course, a president's budget is never wholly adopted by Congress. Both the House and Senate vote on budget proposals, and they often vote on several versions.

Record number of insured children

Currently, a record number of children have some kind of health insurance.

CHIP also has a history of strong bipartisan support. Republican Newt Gingrich's Congress passed the legislation, and it was signed into law by a Democrat, President Bill Clinton. It was co-sponsored by Republicans, including Sen. Orrin Hatch and Sen. Chuck Grassley, and was backed by Democrats like Sen. Ted Kennedy and Rep. Bobby Rush.

Still, it hasn't always received unanimous support. President George W. Bush vetoed an expansion of the program. The current Health and Human Services secretary, Dr. Tom Price, voted twice against expanding CHIP when he was in Congress.

The federal funding costs taxpayers about $15 billion a year . This budget has not been re-authorized by Congress. A congressional committee was supposed to hold a meeting about it in May, but it was abruptly canceled. The money for the program runs out September 30.

The current CHIP "situation," even before the Trump budget announcement, has frustrated governors and other leaders who work with the program.

The National Governors Association (PDF) sent a letter to Congress in May urging it to grant a five-year extension on CHIP funding. It characterized the program as "widely supported by governors, who recognize that access to health insurance is critical to ensuring a healthy start for our nation's children."

"Just over 95% of kids are covered. It is the highest it's ever been, and we don't want to take any steps back," said Lisa Shapiro , vice president of health policy with First Focus, a nonpartisan advocacy organization for children and family policy issues. "We are super disappointed with the whole situation surrounding CHIP. We don't want to be playing with that success."

'States need time to plan'

The delay in funding has been a problem for states. Most state legislatures have wrapped up for the year, and it's hard to plan programs when states don't know what the budgets will be.

Some states may be able to fund CHIP through 2018, regardless of federal funds, according to the National Governors Association, but all states will exhaust their federal allotments at some point in fiscal year 2018. That means families could be getting letters as early as this summer telling them to make other arrangements for their child's health care, if they can afford other options.

"It's not like a light switch that can turn on and off. States need time to plan and notify families," Shapiro said. "This creates a lot of uncertainty for the families who are enrolled."

The Medicaid and CHIP Payment and Access Commission, a nonpartisan group (PDF) appointed by the US comptroller general that gives advice to Congress, told lawmakers they should extend the budget for additional time. The commission suggested that the "urgency of congressional action to preserve health insurance coverage for the nation's low- and moderate-income families cannot be overstated," according to the letter sent by Chairwoman Sara Rosenbaum.

Maureen Hensley-Quinn, a senior program director with the National Academy for State Health Policy, said it has been working with states since last summer to prepare for whatever may come from Congress. Some states have prepared different notification letters for families to quickly adjust to whatever Congress decides.

"No state -- I feel really comfortable saying this -- no state wants to dis-enroll kids. No state wants to freeze enrollment," Hensley-Quinn said. "They all are really hopeful Congress will pass extended CHIP funding."

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But in the meantime, they have to think concretely about steps to take should federal funding for CHIP, Medicaid or the exchanges shrink or dry up, Hensley-Quinn said.