WASHINGTON — About a week after the first report of a Covid-19 case at a meatpacking plant in southwest Kansas in early April, the state’s governor, Laura Kelly, issued a pointed warning to President Trump: Without test kits to separate the well from the sick, a fast-moving outbreak could idle facilities that produce roughly one-quarter of the nation’s meat supply.

Within three days, 80 blue-and-white boxes of test kits and testing machines arrived, and two Black Hawk helicopters from the Kansas National Guard whisked them to the afflicted region. As the test results came in last week, the costs of the delay became clear: 250 workers in six plants were already infected.

In Albany, Ga., a hot spot for the disease, a hospital finally figured out a way to run its own coronavirus tests, rather than relying on limited state capacity or outsourcing the work to slow-moving private labs. But it still struggles to run as many tests as it would like because of a shortage of components.

In Ohio, a research institution in Columbus is teaming up with a plastics company to churn out nasal swabs on 3-D printers for use in the state. But when Mysheika W. Roberts, the city’s health commissioner, offered test kits to local health centers, she learned they lacked the protective gear they needed to put them to use.