SEOUL—North Korea is confronting its worst drought in nearly 40 years, its state media said, as international relief groups warn of a mounting food shortage inside the cash-strapped regime.

The impoverished country of 25 million has long battled hunger. But now 40% of North Koreans face an “urgent need” for food assistance, two U.N. humanitarian groups said earlier this month. The country received just 2.1 inches of rainfall from the beginning of January to early May, the lowest amount for that time period since 1982, KCNA,...