Duke Energy Corp. was ordered two years ago to clean up coal-ash ponds in North Carolina that posed risks to the environment and public health. The company won’t be done in time for Hurricane Florence.

The Category 4 storm bearing down on the Carolinas may deliver flooding rains with the potential to overwhelm the earthen berms containing toxic waste from power plants, according to Frank Holleman, senior attorney at the Southern Environmental Law Center. Florence, with top winds...