In simple, pendulum of the clock moves slow during summer is due to thermal expansion of solids and moves faster during winter is due to contraction of solids.
1. Pendulum of the clock is made up of a metal, will be heated up due to increase in the temperature of the surroundings in summer which leads to expansion of the pendulum (linear expansion), Whereas during winter, the metal contracts due to decrease in the temperature of surroundings. As you know, temperature is directly proportional to length, the length of the metal changes based on the temperature change. Hence, we find a a small gap between the needle and pendulum edge, so that it wont collide due to change in the temperatures.
2. As temperature is inversely proportional to frequency of the vibrations (t=1/f), the pendulum moves slow in summer season and fast in winter season. Hence, we will be setting up the time during winter and summer seasons.