cave

a hollow space under the ground, generally opening into the side of a hill, large enough for people to enter

Adullam

This is the name of a town near the city of Gath. (See: rc://en/ta/man/translate/translate-names)

heard that David had gone to hide in the cave

This is a generalization. AT: "Many people who were distressed ... many people who were unhappy" (See: rc://en/ta/man/translate/figs-hyperbole)

military officer who leads soldiers