Primary results 2022: Georgia, Alabama, Arkansas, Texas runoff and Minnesota special election primary results

Incumbent Republican Gov. Brian Kemp of Georgia is projected to survive a primary challenge by former Sen. David Perdue, who received former President Donald Trump's biggest endorsement and the most financial help he's given to any candidate so far. 

Trump-backed candidates appeared to be foundering Tuesday night in Georgia , although his preferred Senate candidate, Herschel Walker, is projected to win the Republican nomination. But CBS News projected Attorney General Chris Carr survived the challenge from the Trump-backed John Gordon. 

And CBS News projected Secretary of State Brad Raffensperger, whom Trump asked in January 2021 to "find" enough votes to overturn the results, will win the Republican nomination over his challenger, Trump-backed U.S. Rep. Jody Hice. He was one of the House Republicans who challenged the election results in Pennsylvania and Arizona on Jan. 6, 2021. 

"My thinking was the vast majority of Georgians are looking for honest people for elected office. Someone who would do their job, follow the law, look out for them — regardless of the personal cost to do so," Raffensperger said. "Standing for you, standing for the rule of law and election integrity, and standing for the truth — and not buckling under the pressure is what people want. I was elected to serve the people and so I shall. And so I have."

Trump had targeted Georgia Republicans for not overturning the 2020 election results. 

Kemp, meanwhile, will take on Democrat Stacey Abrams, whom he defeated in 2018. 

"I am fully supporting Brian Kemp in his run to beat Stacey Abrams," Perdue told supporters in his concession speech. "We're going to do everything we can to make damn sure Stacey Abrams doesn't take over this state."

Without mentioning Kemp's victory, Trump called into Walker's victory party, touting his "great numbers." Walker noted "the big game is in November," when he faces incumbent Sen. Raphael Warnock. 

In one of the closely-watched House primaries, CBS News projected Rep. Marjorie Taylor Greene will win the Republican nomination in Georgia's conservative 14th District, easily defeating a number of challengers. 

In her victory remarks, Greene issued a stark warning to establishment Republicans and "globalist elites." "Sending me back to Washington will send a message to the blood-sucking establishment. It is we who will set the political agenda for the next decade, and not them," she said.

In Georgia's major Democratic primary, two incumbents faced off in the new 7th District. CBS News projected Rep. Lucy McBath, who was redistricted out of Georgia's 6th District, will defeat Rep. Carolyn Bourdeaux, who represented Georgia's 7th District.

Alabama and Arkansas are also holding primaries Tuesday. Trump has made his preferences known in the Senate race for the open seat in Alabama twice. He first backed longtime loyalist Mo Brooks, but later changed his mind and dropped his endorsement in March, when Brooks was struggling in the polls. After Trump pulled his endorsement, Brooks, who was one of the members of Congress who objected to election results, said the former president "asked me to rescind the 2020 elections."

But despite losing Trump's endorsement, CBS News projected Brooks and Katie Britt, current Sen. Richard Shelby's former chief of staff, would advance to a runoff.  

In Arkansas, CBS News projected Trump's former White House press secretary, Sarah Huckabee Sanders, won the gubernatorial primary, after leading the pack in fundraising and polling. Her father, Mike Huckabee, was Arkansas governor from 1996 until 2007. 

A few crucial runoffs are happening in Texas, which dealt with a shooting Tuesday near San Antonio. In Texas' 28th District, incumbent Rep. Henry Cuellar faces Jessica Cisneros after Cuellar failed to net 50% of the vote in the March 1 primary. As of late Tuesday, fewer than 200 votes separated the two. 

CBS News projected Texas Attorney General Ken Paxton will win the Republican nomination, easily defeating Land Commissioner George P. Bush. While Paxton is mired in many scandals, Bush — the son of former Florida Gov. Jeb Bush, nephew of former President George W. Bush and grandson of former President George H.W. Bush — couldn't overcome his family name.

According to an April poll by the Texas Hispanic Policy Foundation, 40% of Republican primary voters said they would never vote for Bush. Two-thirds of those voters said that's because he is a member of the Bush family. Forty-one percent said they wouldn't vote for Bush because he's not conservative enough.

Bush tweeted a concession and called attention to Tuesday's shooting in Uvalde, Texas, which left at least 19 children and two adults dead. 

"But after the tragic events of earlier today, it's important to keep life's temporary disappointments in perspective," Bush said. "There are grieving parents in South Texas today. My family and I are grieving with them."

Bo Erickson contributed to this story.