No, the distance traveled by an electromagnetic wave in a given period of time does not depend on its frequency. The speed of light, which is the speed at which electromagnetic waves travel in a vacuum, is constant regardless of the frequency of the wave.
In a vacuum, all electromagnetic waves, regardless of their frequency (or wavelength), travel at the same speed, which is approximately 299,792,458 meters per second (or about 186,282 miles per second). This value is often denoted by the symbol 'c' in physics.
The frequency of an electromagnetic wave refers to the number of wave cycles that pass a given point in one second. The wavelength, on the other hand, is the distance between two corresponding points on successive waves, such as between two peaks or two troughs.
While a higher frequency wave indeed has more cycles or oscillations per second compared to a lower frequency wave, each individual cycle or oscillation still covers the same distance as any other cycle of any other frequency wave in a given period of time. It is the wavelength that varies inversely with frequency, meaning higher frequency waves have shorter wavelengths and vice versa.
Therefore, the distance traveled by an electromagnetic wave in a specific period of time, say one second, is the same regardless of the frequency. The only difference is that higher frequency waves have more wave cycles in that time frame, but each individual cycle still covers the same distance as any other cycle of any other frequency wave.