When President Trump told governors that they needed to step up their efforts to secure medical supplies, Gov. Larry Hogan of Maryland took the entreaty seriously and negotiated with suppliers in South Korea to obtain coronavirus test kits.

“The No. 1 problem facing us is lack of testing,” said Mr. Hogan, a Republican, who has been among the many critics of the Trump administration’s repeated claims that states have adequate testing provided by the federal government. “We can’t open up our states without ramping up testing.”

In recent days, his wife, Yumi Hogan, a Korean immigrant who speaks fluent Korean, had been on the phone in the middle of the night helping to secure the final deal with two labs to sell Maryland the tests.

“Luckily we had a very strong relationship with Korea,” Mr. Hogan said. “But it should not have been this difficult.”