“How can I help my 6-year-old blue and gold macaw named Hugo to overcome his fear when I move while he is perched on my hand, despite being comfortable with stepping up and staying on my hand? Additionally, Hugo enjoys head scratches and being touched, and he resides in a spacious cage measuring approximately 5 square meters.”
Try this relaxed training I do with my canary.
Whenever I train my canary, we work on the same step for a week before moving onto the next one.
This requires a lot of patience on your part but I think it's the best way to get a bird fully comfortable with a new behavior.
Speaking to him during training will also make it more comfortable for him.
Always keep his favorite food in your other hand to reward him.
This training works best if there's a perch or nearby object you can carry him to, preferably something close to his cage.
Start by identifying the exact amount of movement he tolerates before getting scared.
Then, when the bird is on your hand, move it side to side as slow as possible. Reward him if he stays on your hand.
Repeat this process consistently for a week.
Once the bird is used to this, ever so slightly increase the amount you move your hand.
Repeat this for another week.
You will do this for several weeks until you are closer and closer to your desired object. Once you are fully able to carry the bird to the object is when you start speeding up the movement.
It will take some time, but try it and tell me how it goes.