No, the photon would still travel at the speed of light, even when fired from the back to the front of a tube moving at 1/10th the speed of light. This is due to the principles of special relativity.
According to the theory of special relativity, the speed of light in a vacuum is constant for all observers, regardless of their relative motion. This means that the speed of light is the same in all inertial reference frames.
In the scenario you described, when the tube is moving at 1/10th the speed of light, an observer inside the tube and another observer outside the tube would both measure the speed of light to be the same. From the perspective of an observer inside the tube, the photon would still travel at the speed of light. From the perspective of an observer outside the tube, the photon would also appear to be moving at the speed of light, not 110% the speed of light.
This is because the effects of time dilation and length contraction come into play when objects are moving relative to each other at high speeds. These effects ensure that the measured speed of light remains constant for all observers, regardless of their relative motion.
Therefore, even in a scenario where the tube is moving at a significant fraction of the speed of light, the photon will still travel at the speed of light relative to any observer. It would not exceed or surpass the speed of light, as that is not permitted by the laws of physics as we currently understand them.