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In a wide-ranging speech on “ending systemic racism,” Hillary Clinton presented a $125 billion plan to assist poor and minority communities with job training, education and re-entering society after incarceration, part of an effort to speak directly to African-American voters as the Democratic primary contest heads to South Carolina.

Her remarks, delivered in Harlem, were centered around what Mrs. Clinton called a “Breaking Down Barriers” agenda that would disproportionately help in “places where people of color and the poor have been left out and left behind.” And she particularly named “places like Harlem and rural South Carolina.”

Without mentioning her opponent, Senator Bernie Sanders of Vermont, who also met with civil rights leaders on Tuesday, Mrs. Clinton implied that his central message of taking on Wall Street and economic inequality did little to address problems such as gun violence, joblessness and clashes with the police that disproportionately affect black communities.

“Just imagine with me, imagine if white kids were 500 percent more likely to die from asthma than black kids,” she said. “Imagine if a white baby in South Carolina were twice as likely to die before her first birthday than an African-American baby.”

“These are not only problems of economic inequality,” Mrs. Clinton continued. “These are problems of racial inequity, and we need to say that loudly and clearly.”

Later in the speech, Mrs. Clinton emphasized the deep relationships that she and her husband, Bill Clinton, have formed with civil rights leaders, from her early work for Marian Wright Edelman, the founder of the Children’s Defense Fund.

Her voice going hoarse, and with the crowd cheering her on with chants of “Hillary!” Mrs. Clinton quietly uttered the most stinging line of the speech against Mr. Sanders’s efforts to reach out to black voters: “You can’t start building relationships a few weeks before a vote.”

Mrs. Clinton’s plan, details of which were already released, includes a $20 billion program to create jobs for young people; provide funding for local programs that would end what she called the “school to prison pipeline”; a $5 billion investment to fund re-entry programs that would help formerly incarcerated people move back into society; and $25 billion to aid entrepreneurs in underserved communities.

A campaign aide said the plan would be paid for by a “risk fee” imposed on Wall Street banks and other changes in the tax code.

But Mrs. Clinton did not get bogged down in policy plans, and her speech, held in the Langston Hughes Auditorium at the Schomburg Center for Research in Black Culture on Malcolm X Boulevard, went beyond the anti-racism agenda to include a litany of problems that affect black communities, from voting rights to an overhaul of the criminal justice system, as well as criticism of President Obama.

She called on white Americans to be more empathetic to the problems that plague black communities, including the water crisis in Flint, Mich. And she implored Republicans to respect Mr. Obama’s right to nominate a replacement for the late Justice Antonin Scalia on the Supreme Court.

“Many Republicans talk in coded racial language,” she said to applause from the audience. “They demonize President Obama and encourage the ugliest impulses of the fringe” of the party.

“The president has the right to nominate under the Constitution,” she said. “And the Senate has the obligation to process that nomination.”

Mrs. Clinton was accompanied on stage by some of New York’s most prominent Democrats, including Gov. Andrew Cuomo and his partner, Sandra Lee; Mayor Bill de Blasio and his wife, Chirlane McCray; Eric H. Holder Jr., the former attorney general; and Representative Charles B. Rangel of New York, a tableau that underscored Mrs. Clinton’s deep support among party leaders.

In introducing Mrs. Clinton, Mr. Rangel, who represents Harlem, did not mention Mr. Sanders but he emphasized that the political momentum would soon turn back in the former first lady’s favor. “Remember today,” Mr. Rangel told the audience, “We can tell our children and grandchildren, ‘We were there when she turned it around.’”