The bud might be a growth from your pulp called as pulp polyp if it is growing within the grossly decayed tooth. If it growing adjacent to it may be it's an abscess filled with pus due to infection from dental caries. You need to visit a dentist n get better diagnosis. Usually it involves either extraction or root canal treatment followed by cap

This could be pus formation or a small tumor. The decayed tooth has to be treated immediately. If it can be saved by doing a root canal treatment, well and good. If it has crossed that stage, then it has to be removed. Clinical evaluation and x-rays are required before the definite treatment can be decided.

it's pulp polyp, a kind of tumor ,, you need to extract the tooth or perform root canal therapy whichever is possible ,, need for x-ray, donot bite on flesh you might bleed and get infected..

Although it cannot be said for sure based on the description that you have mentioned, but it seems that you have granuloma developing alongside the decayed tooth..... extraction of such tooth it will be recommended if root canal treatment is not possible and this can be judged only after an xray. Regards