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The hard pus from the small bump is the collected cell debris.
This condition is localised acne
and it should be treated with local application of keratolytic
creams twice a day. these creams loosen the oily texture of the skin so the gland there open up and released the secretions which currently are getting piled up and infected.
You should wash the face twice a day with a medicated soap and mop dry the face with thick cotton towel immediately. apply keratolytics and apply antibiotic cream after two hours to follow to clear the established infections.
AVOID FATTY FRIED AND FAST FOODS AND CUT DOWN ON THE NONVEG IF ANY. REDUCING BODY WEIGHT WILL ALSO HELP.
Go for a brisk walk or a jogging every day to sweat a little so the skin gets toileted from inside.
if still the trouble remains more than a months then you should visit a local dermatologist
I hope the advise would be informative and useful for you.