One of the main reasons a blackhead can become infected is by touching our skin with dirty fingers.
Many times people try to pop out a blackhead without washing their hands and face. We carry a lot of germs and bacteria on our fingers, when touching our face with dirty hands, bacteria can enter pores causing an infection.
The best way to deal with an infected blackhead is to apply a strong spot solution, high in salicylic acid, and leave it on overnight. This should dry out the infected blackhead. "Bye Bye Blemish" is one of the best overnight solutions and it is not very expensive, you can purchase it at the beauty supply store, or online. Or you can make your own home solution. Dab cotton swab in a freshly squeezed out lime juice and apply it directly to the infected blackhead. It might not work as fast as the over the counter solution but it doesn't have any chemicals, which is always great for your skin.