There have been a number of movies about damage to the moon, but I'll suggest just one right now:
Earthstorm (2006 TV movie)
starring Stephen Baldwin, Amy Price-Francis, Dirk Benedict
A massive asteroid impact on the moon begins causing storms on earth due to the sudden changes in ocean tides. But when further examination is conducted it's discovered that the moon's structure is now entirely unstable -- threatening all life on earth. With time running out, a team of scientists turn to one man, demolitions expert John Redding, in effort to find a solution and secure the moon.
Here's another possibility:
Impact (2009 TV miniseries)
starring David James Elliott, Natasha Henstridge, James Cromwell, Steven Culp
While the entire world watches the largest meteor shower in 10,000 years, a rogue asteroid, hidden by the meteor field, smashes into the moon in a tremendous explosion of rock and debris. Fragments from the asteroid and even from the moon itself penetrate Earth's atmosphere and make impact. Even though the initial damage is minimal, there is significant physical damage to the lunar surface. Experts quickly conclude there will be no lasting ramifications. Then strange anomalies begin to manifest themselves on Earth: cell phone disruptions, unusual static charges and odd tidal behavior. The world's leading scientists begin piecing together evidence that suggests the moon's properties and its orbit may have been permanently altered.