That hasn't been possible.
A grand jury is investigating possible crimes connected with the affair. Lawmakers from both sides of the aisle have called for the Republican governor to resign, and his standing with the legislature, which is dominated by his own party, appears to have been permanently weakened. Some Republicans are even blaming him for the loss of a state House seat in a special election on Tuesday.
Scott Dieckhaus, a former executive director of the Missouri House Republican Campaign Committee, tweeted Wednesday morning:
I am going to say what no one else is right now - we (Missouri Republicans) just lost a MO House seat because of @EricGreitens and his refusal to put our party over his ego. @RobertKnodell and the @Missouri_HRCC team ran a hell of a race and deserved a win. — Scott Dieckhaus (@sdieckhaus) February 7, 2018
It's his top legislative priority for the year, but it appears to be dead on arrival. Legislators don't seem inclined to take up much of anything that the governor has to offer.
"He doesn't have a whole lot of support in the legislature that I can tell," says Nate Walker, one of five House Republicans who has called on Greitens to step down.
Walker was an early supporter of Greitens, a former Navy SEAL who overcame a large field in the 2016 GOP gubernatorial primary to win his first campaign for public office. Greitens' background made him look like a potential star, one who appeared in an ad touting neighboring Illinois Gov. Bruce Rauner.
Greitens never got out of campaign mode. He ran as an outsider vowing to clean up Jefferson City, and he never let up his attacks after arriving there. He called out individual legislators on social media, and outside groups affiliated with the governor ran negative ads against those who crossed him. Greitens quickly signed a bill turning Missouri into a right-to-work state, but his own priority legislation rarely got far.
That was last year, before his affair became public. Now that he's in deeper trouble, Republicans appear to be even less inclined to support the governor's agenda.
"His relationship with the GOP-dominated state legislature is fractured and there appears to be little desire on the part of legislators to repair it," says Peverill Squire, a political scientist at the University of Missouri. "At this point, the governor is without many friends in Jefferson City."
Lawmakers are nervous about what further news might break regarding his affair. The woman's husband secretly tape-recorded her saying that Greitens had threatened to release a compromising photo of her if she went public. The husband's attorney said Monday that he had been asked to testify before a grand jury investigating the possible blackmail scheme, which Greitens has denied. (He has not specifically confirmed or denied taking a picture of the woman.)
"Now a grand jury has been empanelled and it will be a drip, drip, drip," Rep. Walker says. "Lots more is going to come. I've been around long enough to observe or watch these things happen."
Greitens' troubles come at a time when allegations of sexual misconduct have ended the careers of numerous politicians around the country.
In his successful state House race in this week's special election, Democrat Mike Revis featured damaging headlines about Greitens in a web ad. Revis carried a Jefferson County district where President Trump had taken 61 percent of the vote in 2016. Republicans held the three other House seats at stake on Tuesday, but the Democratic candidates outperformed Hillary Clinton's 2016 showing in the districts by margins of 18, 25 and 53 percentage points.
"Certainly, Gov. Greitens was a factor," says Democratic state Sen. Jill Schupp. "I think Republicans are doing everything they can to distance themselves from the governor, and I don't blame them."
Nothing hurts a politician's standing like the sense that he's a drag on the party as a whole. Perhaps Greitens' troubles will clear up by November, but Missouri Republicans aren't yet ready to bet on it.
"He's damaged," says David Robertson, who chairs the political science department at the University of Missouri-St. Louis. "Maybe he can ride it out, but his political future is a whole lot dimmer than it used to be."