With 0 replies it won't hurt to offer one.

I don't know the material you speak of. However as a materials scientist here's what I would do with sticky stuff stuck to a woody material.

1. Get it as cold as possible and it should pop off.

2. #1 is your best bet but if it needs help then using a slow evaporating solvent will help. DMSO is very slow evaporating and cheap. The only concern is that any solvent might move the color from the black to the pod. Consider using a syringe with a needle (blunt needle) and use only a drop at the junction of the black stuff and the pod. This will mitigate color movement to the pod. Then after some time try to remove the material.

3. There must be other good ideas if it's a common problem.

If you can find out the official name of the black part of the plant that is a keyword that you must use when you google for answers. Finding that word for the part of the plant might be found from a search about the 'botony of poppy plants' on google.

I hope #1 works for you.