Yes, you need to cut the damaged tip off. Some have known to dry and fall off but if it were me, I would cut it off. And yes, they will generally branch out when you cut them but some have been known to only give one branch when cut. Hope this helps.

I would also seal the top of the branch after you cut and the latex has dried ( about an hour). This will prrevent fungus etc...Some use Dap sealant or Elmer's waterproof wood glue. Seal entire surface of the new cut stem....you should see it back bud this spring or summer!

Cut until you see clean healthy wood... ( use clean tools wiped with alcohol)

Good luck!!

Laura