On his social media platform, Mr. Trump wrote on Nov. 11 that his endorsement of Mr. Youngkin had cemented the governor’s 2021 victory. Mr. Youngkin is little known to Republican voters beyond Virginia, barely registering in early primary polls. Supporters envision Mr. Youngkin winning over droves of primary voters in the intimate campaign settings of Iowa and New Hampshire. But now, Mr. Centers said, Mr. Youngkin needs to build on that victory. “They didn’t want the other students to feel bad,” Mr. Youngkin said.