Could an asteroid impact push the Moon closer to us?
The Moon is very big, and any small object hitting it would have very little effect on its motion around the Earth, because the Moon’s own momentum would overwhelm that of the impact. Most asteroid collisions would result in large craters and little else; even the largest asteroid known, Ceres, wouldn’t budge the Moon.
However, if an object of similar mass and velocity to the Moon were to hit it, the Moon’s orbit could well be altered, though it’s more likely the Moon would be destroyed by such an impact. If the Moon were to orbit closer to Earth we would experience much larger tides, along with longer and more frequent solar eclipses.