[nine_k]

The amount of energy required to lower the orbits of enough asteroids (e.g. 20% of Mars total mass) is probably enough to melt a significant portion of Mars. Unfortunately, collisions would mostly heat up the surface, not the core. Even if you could precisely deorbit enough asteroids to hit Mars, the results of the bombardment would probably take a long time to settle.

Planetary-scale engineering is not only hard, it's inevitably slow.

I think cleaning up the atmosphere of Venus by spreading CO2-eating anaerobic air-suspended algae is more realistic, and the results would likely be nicer.