How much the moon movement can influence an equatorial mount setup? If I use an equatorial mount for my telescope to view the moon, is there a way to take in count the moon movement around the earth, or the movement is too small compared to earth rotation?
For most purposes, the difference between the Moon's motion across the sky and star's motion (the sidereal rate) is negligible. The Moon appears to move about 1/2 degree per hour relative to the stars, so while the stars travel 15 degrees per hour east to west, the Moon travels approximately 14.5 degrees per hour. Unless your mount is aligned accurately and has an accurate motor, you probably will not notice this difference.

Some telescope drives do have a "lunar rate" so that it will track the Moon more accurately. But these approximations do not take into account the fact that the Moon is also moving north or south in declination as it orbits the Earth; the lunar rate only compensates for its west to east motion. (Maybe a goto scope will track the Moon more accurately since it theoretically could be calculating the position of the Moon and pointing to a changing position.)