I am not sure if there's a word/expression/phrase that describes that cute pose some dogs get into to get favors. I am not even sure if dogs do get into those kind of poses, correct me if I am wrong. The only word or expression that comes to mind is cutie pose, but I doubt it means what I think it means. It's a general expression and certainly it doesn't refer to that exact pose, but I couldn't think of anything else.
When I have trained my dogs, I usually give the command "beg" for the pose you describe while offering a treat - a small biscuit or equivalent.
So I would use the word "begging".