This will lead to mastitis--an inflammation and infection of the nipples. Mastitis occurs commonly when dogs are nursing. Mastitis can be very painful and needs to be treated. I recommend having a veterinarian examine your dog and start appropriate treatment.
Your vet needs to prescribe something because it is getting infected. The est way to loosen the clogged nipple is to use warm water compresses. Take a wash cloth and a bowl of warm water and apply the compresses four or five times a day. It is really important to take your little dog to the vet also.