According to our current understanding of physics, galaxies cannot move away from the Milky Way (or any other galaxy) at speeds greater than the speed of light. The expansion of the universe, as described by the theory of cosmic inflation and confirmed by observations of cosmic microwave background radiation, causes galaxies to recede from each other. However, this expansion does not involve galaxies moving through space at speeds exceeding the speed of light.
If a galaxy were hypothetically moving away from the Milky Way at twice the speed of light, it would be beyond our ability to observe or detect directly. According to Einstein's theory of special relativity, as an object with mass approaches or exceeds the speed of light, its relativistic mass becomes infinite, and it would require an infinite amount of energy to accelerate it to or beyond that speed. This means that such a hypothetical galaxy would lie outside our observable universe.
In terms of gravitational influence, according to general relativity, gravitational effects propagate at the speed of light. Therefore, if a galaxy were located beyond our observable universe, its gravitational influence on the Milky Way would not be directly observable or tangible.
It's important to note that our current understanding of the universe is based on well-established scientific theories and observations. The speed of light is a fundamental limit within our current understanding of physics, and objects moving faster than light are not consistent with our understanding of the laws of physics.