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Seth Wenig, Associated Press
Congressman Charles Rangel is followed by reporters in New York, Monday, June 23, 2014. With Primary Day just a day away, longtime congressman Rangel says he's not overconfident he'll prevail over his closet rival — but insisted that voters would stick with a veteran lawmaker, dismissing criticism that he was too old to continue serving in Washington.

WASHINGTON — Six-term Sen. Thad Cochran struggled against a challenge from state Sen. Chris McDaniel in Mississippi's Republican primary runoff Tuesday, testing the appeal of decades of delivering federal largesse against conservative demands to slash government spending.

The 76-year-old Cochran, the former Senate Appropriations Committee chairman who has steered billions of dollars back home, was locked in a tight race against the tea party-backed McDaniel, who was born the year Cochran came to Washington as a congressman. When no clear winner emerged from the June 3 primary, it forced a three-week campaign dash for the nomination.

"I think it's very close, very doable," Sen. Roger Wicker, R-Miss., said late Monday after campaigning for Cochran.

McDaniel, in a swipe at Cochran and John McCain of Arizona, a Cochran supporter, criticized "outrageous spending sprees" by the two establishment senators and pledged to stand with conservatives like Sens. Ted Cruz of Texas and Mike Lee of Utah.

The runoff is the headline race on a busy day of primaries in New York, Oklahoma, Maryland, Colorado and Utah. A special election in Florida will fill the U.S. House seat once held by Rep. Trey Radel, who resigned in January after pleading guilty to cocaine possession.

In New York, Democratic Rep. Charles Rangel, 84, who survived House censure in 2010 over financial wrongdoing, was seeking a 23rd term. State Sen. Adriano Espaillat, a Dominican-born legislator who lost by fewer than 1,100 votes in the 2012 primary, poses the toughest challenge in the majority Hispanic district that includes Harlem.

In closing arguments in Mississippi, Cochran backers portrayed him as experienced and a reliable ally for veterans and the military.

"We are facing a crisis with our veterans. We are facing a crisis internationally," McCain said after an appearance on Cochran's behalf. "His opponent has no experience or knowledge with those issues."

That pitch seemed to sway Shelia Stokley, a 55-year-old pharmacy technician who cast her ballot for Cochran Tuesday at the National Guard Amory in Morton.

"I just think he's the one we need for right now," said Stokley, 55.

Officials said more absentee ballots had been requested for Tuesday's elections than for the June 3 first round of voting, suggesting turnout might be heavier.

Outside groups, from tea party organizations to the U.S. Chamber of Commerce, have spent some $12 million on the race. Cochran appealed to Democratic voters — white and black — to cross over and back him, prompting tea partyers to promise to send observers to the precincts.

Conservative purists and the tea party movement, born five years ago in the fight over federal government spending and President Barack Obama's health care law, have had more losses than wins this election cycle, although the triumphs have rattled Washington.

Mainstream conservatives have turned back challenges in high-profile Senate and House races in Kentucky, North Carolina, Georgia and Idaho. Tea partyers prevailed in Texas' lieutenant governor race. Conservatives opposed to immigration reform scored a resounding win in Virginia, where little-known college professor Dave Brat knocked out House Majority Leader Eric Cantor in the Republican primary on June 10.

The Mississippi runoff was one of several internecine GOP contests.

In Oklahoma's Senate primary, two-term Rep. James Lankford, a member of the House Republican leadership, was battling T.W. Shannon, who was the state's first black House speaker and is a member of the Chickasaw Nation. National tea party groups and the Senate Conservatives Fund have backed Shannon, who also has the support of Sarah Palin and Cruz.

The winner in the solidly Republican state will replace Sen. Tom Coburn, who was diagnosed with a recurrence of prostate cancer. Coburn said a desire to focus on other issues, not his health, was the reason he was retiring with two years left in his term.

Lankford and Shannon could emerge from the crowded primary as the two candidates in an Aug. 26 runoff.

National Republicans are nervously eyeing Colorado's four-way gubernatorial primary, which includes 2008 presidential candidate Tom Tancredo, an opponent of immigration whose presence at the top of the ticket could undercut GOP prospects in November's Senate and House races.

Tancredo faces former Rep. Bob Beauprez, Colorado Secretary of State Scott Gessler and former state Sen. Mike Kopp for the right to challenge Democratic Gov. John Hickenlooper, who is given the edge.

In Florida, voters in a strongly Republican district were choosing a replacement for Radel. Republican Curt Clawson, a former CEO of an aluminum wheel company, was favored to win against Democrat April Freeman and Libertarian Ray Netherwood.

Associated Press writer Jack Elliott in Morton, Mississippi, and Philip Elliott in Washington contributed to this report.