The Note's Must-Reads are a round-up of today's political headlines and stories from ABC News and the top U.S. newspapers. Posted Monday through Friday right here at www.abcnews.com

Compiled by ABC News' Carrie Halperin, Jayce Henderson, Will Cantine and J.P. Lawrence

ANTHONY WEINER The Washington Post's Karen Tumulty: " Weiner Admits More Lewd Conduct, Vows To Stay In New York Mayoral Race" Anthony Weiner, the disgraced former congressman who polls suggest is a leading candidate for mayor of New York, admitted Tuesday that he engaged in a series of sexually explicit communications with a young woman on the Internet. The behavior allegedly occurred in 2012, more than a year after Weiner was forced to resign from Congress after admitting to similar episodes. But in a news conference that seemed a little like a public therapy session, Weiner stood with his wife, Huma Abedin, and vowed to stay in the race for mayor. LINK

Politico's Glenn Thrush: " Huma And Anthony: Scenes From A Marriage" New York City Mayoral candidate Anthony Weiner unleashed his typical torrent of verbiage on Tuesday - but words couldn't compete with the excruciating image of his embarrassed wife squirming in the spotlight both have sought. In the most cringe-worthy moment of a four-month political comeback yarn that has tested Gotham's gag reflex, Huma Abedin, the dignified wife of a serially disgraced Democrat, nervously defended her husband at a Tuesday press conference. LINK

U.S. DEBT LIMIT / ECONOMY Bloomberg's Roxana Tiron: " Boehner Signals Clash With White House on U.S. Debt Limit" House Speaker John Boehner signaled a clash with the White House and the Democratic-led Senate over raising the U.S. borrowing authority later this year. Congressional Republicans are staking their ground in fiscal negotiations that once again could pose the threat of default or a government shutdown - the recurring theme surrounding efforts to reduce the nation's deficit since 2011. LINK

The Washington Times' Dave Boyer: " Economic progress mixed despite Obama claims" President Obama will take to the road to justify his new economic plans Wednesday based on what aides call the "significant" progress of the past five years, but economic benchmarks during his presidency show a mixed bag of gains and setbacks on pocketbook issues. LINK

USA Today's Gary Strauss: " Could door-to-door Postal Service delivery end soon?" First, it was doing away with Saturday delivery. Now, door-to-door service could be coming to an end. In an effort designed to cut costs at the cash-strapped agency by up to $4.5 billion a year, Congressman Darrell Issa, R-Calif., is proposing the U.S Postal Service phase out door-to-door delivery and shift service curbside and to neighborhood cluster boxes. The proposal - due for vote by the House Oversight and Government Reform Committee on Wednesday - would affect about 37 million residences and businesses. LINK

CONGRESS ABC News' Matthew Larotonda: " Harry Reid: Tea Party 'Obsessed' House Endangering Major Transportation and Housing Bill" The Senate has agreed to move forward with a measure worth $54 billion in funding for transportation and housing projects, but the bill's future is far from clear. The bill, marking fiscal year 2014 appropriations for HUD, the Department of Transportation, and related agencies, passed a major procedural hurdle today when the Senate voted 73-26 to continue consideration of the legislation. It will now be up for a final simple vote, possibly next week. LINK

The Wall Street Journal's Neil King J. and Rebecca Ballhaus: " Approval of Obama, Congress Falls in New Poll" Rising frustration over partisan gridlock in Washington has damaged faith both in President Barack Obama and lawmakers on Capitol Hill, with disapproval of Congress hitting an all-time high, a new Wall Street Journal/NBC News poll found.The public sourness comes just as Congress and the White House on Tuesday gave new indications they are headed this fall for the kind of clash over federal spending that has in the past driven up public disapproval. In the latest poll, Mr. Obama's job-approval rating fell to 45%, its lowest level since late 2011, while overall disapproval of Congress was at 83%, the highest level ever for Journal polling. LINK

The New York Times' Jonathan Weisman: " House G.O.P. Sets New Offensive On Obama Goals" Congressional Republicans are moving to gut many of President Obama's top priorities with the sharpest spending cuts in a generation and a new push to hold government financing hostage unless the president's signature health care law is stripped of money this fall. As Mr. Obama prepares to deliver a major economic address on Wednesday in Illinois, Republicans in Washington are delivering blow after blow to programs he will promote as vital to a more robust economic recovery and a firmer economic future - from spending on infrastructure and health care to beefing up regulatory agencies. While Mr. Obama would like to keep the economic conversation lofty, his adversaries in Congress are already fighting in the trenches. LINK

PRESIDENT OBAMA The Boston Globe: " Obama is right: US needs broader discussion on race" It's only a slight exaggeration to say that every black American and every white American knows, in their hearts, the truth of President Obama's extraordinary comments on race in the wake of the Trayvon Martin case. Perhaps somewhere there's a place where black men don't have "the experience of walking across the street and hearing the locks click on the doors of cars," as Obama described it; there may also be places where white women never clutch their purses when alone in an elevator with an unfamiliar young black man. But these patterns of learned responses, their causes and effects, should be understandable to almost all black people and white people. LINK

BOOKMARKS The Note: LINK The Must-Reads Online: LINK Top Line Webcast (12noon EST M-F): LINK ABC News Politics: LINK George's Bottom Line (George Stephanopoulos): LINK Follow ABC News on Twitter: LINK ABC News Mobile: LINK ABC News app on your iPhone/iPod Touch/iPad: LINK