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.
Hello,
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.