I have had a cast on my right wrist, distal radius for just over 5 weeks. Can I have it taken off earlier if I take good care of my wrist as I am traveling?

Any fracture? You have a cast on the wrist but does not say what you have it on for? Bones normally heal in 6 weeks if they are fractured. However healing can be variable at your age . A Xray might be needed to see if the area healed. More information needed.