I believe a body falling in toward the sun, by the time it reaches earth, could be doing around 40 km/sec or more. That means 144,000km/hr or less than 3 hours to reach earth from moon orbit. This is barely enough time to launch something, let alone hit it in time to seriously deflect it.
The further away you manage to deflect to path of the object, the more the path diverges for when it reaches the neighbourhood of earth. The biggest problem is the square-cube law. We can barely launch something the size of a space shuttle to earth orbit. Launching a craft to the orbit of Mars or further requires a lot of planning and effort - and if the target is too big, the projectile will not be heavy enoiugh to make a difference. Something a kilometer in diameter, we’re not going to deflect by kinetic force. Presumably, the obvious solution is to pack a load of nuclear explosives and hope they can deflect the body