First of all pls stop pulling it with tweezers and also stop squeezing it.You will only be inviting infection and complicating the picture.From the symptoms it sounds like a sebaceous cyst but without examination,it is very difficult to confirm a diagnosis.It can also be a benign skin growth.I would suggest a dermatologist’s examination and a biopsy of the lump to confirm the diagnosis.
I hope that helps.Take care and pls do keep me posted on how you are doing or if you have any additional queries.Kind regards.
I had what I think was a cyst, and if it was similar to this, here is what worked for me. For about a week straight I refrained from touching it at all, and every night I would take a nice, warm shower or bath to let it soak. After about a week, after letting it soak in the warm water, it had turned white and so I squeezed it firmly until an object fitting your description of the core like thing above popped out. I then cleaned it with rubbing alcohol, put a some neosporin and a bandage on it and it went away just fine.