It is hard to tell what the bumps are without seeing them. The presence of pus usually indicates bacterial inflammation.
This is probably the case with you, too. The dimensions of the bumps might have been helpful. The black spot after popping them might be a small drop of dry blood. The inflammation is accompanied with redness, swelling, and pain of the affected skin. Different parts of the skin might be affected (skin glands, hair follicles or the skin tissue itself).
You might want to see a dermatologist for a physical examination. You might be advised to take local and/or system antibiotics to heal the infection.
Treating the lesions left after the bumps might be done with antiseptics which would keep the lesions clean and certain epithelization products which would speed up the healing process.
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