Both your turning and your English are amazing!!!
Thank you for participating in IAP!!!!
It seems that you know you can straighten the mandrel, but just in case you don't--take all the bushings and blanks off of the mandrel, but bring up the tailstock to support the end. Slowly bring a marker (felt tip if it is available) up to the spinning mandrel, using the tool rest to support the marker until it just touches the mandrel. This will leave a mark on one side of the mandrel. Stop the lathe and tap that side (gently) a couple times. This will bend it back to a straight bar, if done often enough.
Your pens look very good---Please inform us if we are suggesting materials that are not available to you!!
Welcome!