Yes, a stretched string can have kinetic energy. When a string is stretched and then released, it can vibrate or oscillate back and forth. During this motion, the string possesses both potential energy and kinetic energy.
Initially, when the string is at rest in its equilibrium position, it has potential energy due to the tension applied to it. As the string is displaced from its equilibrium position and released, it starts to oscillate. At any point during this oscillation, the string has potential energy due to its displacement from the equilibrium position, and it also has kinetic energy associated with its motion.
As the string moves, its potential energy is converted into kinetic energy and vice versa. At the extremes of its motion, when the string reaches maximum displacement, it momentarily stops and changes direction. At these points, the kinetic energy is at its maximum, while the potential energy is zero. As the string returns to its equilibrium position, the kinetic energy decreases, and the potential energy increases.
So, a stretched string can possess both potential energy and kinetic energy as it oscillates. The energy continuously transforms between the two forms during the vibrational motion of the string.