If one fictitious probe mission was to impact an asteroid fast enough to destroy it using its kinetic energy, could the asteroid's trajectory be predicted accurately enough (plumes or emanations or external forces may change its course) for the probe to hit it exactly where it should (aligned with center of mass)? If not, what kind of trajectory correction system would be required?
It might be, but more likely one would do some kind of a correction burn on route.
When we are going to a far away body, such as Pluto, a KBO, or asteroid, guidance from Earth will only get it so far. Eventually the spacecraft will start to image the target body, and use those images to better map the trajectory it will take. They all have a slight amount of ability to divert, which would allow it to do a course correction and hit the correct spot. Most of the time, these go through Earth, but not all of the time. AIM is an upcoming mission that is very similar to what you want, and Deep Impact is a previous one that worked before.
A couple of missions that have done similar things are Deep Space One and New Horizons.
There is a nice chart from this answer that shows the accuracy vs time of an asteroid. With radar we can get close, but the uncertainty is still quite a bit further then the size of the asteroid.