Are you sure it's stuck, Mary, or could it just be molting? They get real still when they do that. I'd leave it alone, at least for a while.
It's probably either a snowberry clearwing or a hummingbird clearwing. They're the only two hornworms/sphinx moths I know of that use honeysuckle.
Congratulations, assuming your cat is okay! There's nothing cuter than those hummingbird moths!
Sherry

Not certain it's stuck but look at the stem in the photo where it's little end is attached. There is a black spot there and looks like it is also on the body of the little cat. Perhaps they secret something onto the stem to hold themselves in place when they molt? His body is stretched from the stem over to the leaf the other end of his body is attached to. There is nothing to support his body between the stem and the leaf.

I do have hummingbird moths that frequent the nectar plants.

Thank you for your help.

~Mary

What an interesting caterpillar, Mary. Do you have an update on its condition?

He's still alive this morning. By early evening, he was looking worse. His (at least what I think is his head) had shrunk and turned dark, almost black. It wasn't a molt, just looked like he might have a virus or something attacking his body. I felt he was dying so I gently unstuck his backside from the stem and put some leaves over him so the birds wouldn't see him. This morning he had moved his little body more under the leaf he had been feeding on; his head is now the same green color as his body and he looks much better. Other than what I did last night, I've just left him alone.

~Mary

Keep a sharp eye on the little cat. Maybe you should bring it indoors, place in a jar with some leaves for captive rearing. It could be parasitized. It is NOT normal to pass such runny matter that it dries and sticks itself to the plant like that. It is not molting.
Hemaris diffinis.