No, the photon would not be traveling through space at 110% the speed of light. According to the principles of special relativity, the speed of light in a vacuum is always constant and is the same for all observers, regardless of their relative motion.
In the scenario you described, where a tube is moving at 1/10th the speed of light and a photon is fired from the back to the front end of the tube, the speed of light is still measured as 'c' by all observers within the tube. This is because the speed of light is independent of the motion of its source or the observer.
From the perspective of an observer inside the tube, the photon would still travel at the speed of light (c) relative to them. However, an observer outside the tube, who is not moving at 1/10th the speed of light, would also measure the speed of light as 'c' relative to them, even for the photon traveling through the moving tube.
This phenomenon is known as the principle of relativity, which states that the laws of physics, including the constancy of the speed of light, are the same for all inertial observers, regardless of their relative motion. It prevents objects or signals from exceeding the speed of light, regardless of their motion within a moving frame of reference.