Recently I read that Earth's magnetic pole moves quite significantly, several kilometers. Do these changes affect aviation, i.e. for pilots landing their airplanes?
Not really. Yes, there are changes, but they are small, regarding the size of earth. So sometimes they have to renumber runway designations, as they roughly represent the magnetic runway heading. And you get updated maps and airfield charts all the time, where the variation is adjusted - among all the other stuff, that changes like frequencies, obstacles, noise abatement procedures etc.
If you live close to the poles, it might affect you harder, but luckily this regions are not that densely populated. I'm from central Europe, here the variation changes about a half degree per year. You don't actually recognize it, considering all the whisky-compasses used in planes are barely usable to hold a 5 degree heading...
Landings are mostly unaffected, as you either use ground based systems like ILS, satellite navigation or make a visual approach. You should croscheck the runway designator with your actual heading to avoid landing in the wrong direction or on wrong runway. But this is not affected by the change of the poles positions. Ignoring a sometimes proposed complete pole flipover, at least...
I don't think magnetic shifting affects the pilots directly, but it does force airports to re-number their runways because of the fact that runways derive their numbers from their magnetic headings.
Other than that, I don't know of any direct effect this would have on the landing itself, as approaches rely on the ILS's radio signals or direct visual contact with the runway.
While the position of the pole does wander by a bit, this doesn't cause major problems for aviation for several reasons.
The motion of the pole by a few kilometers sounds dramatic. But given the size of the earth represents fairly small changes in the orientation of the magnetic field. Away from the poles, these changes are much harder to detect.
Also, the changes that you mention are not sudden. They're normally measured over years. Lots of other things change in that period of time as well (like runway configurations, changes in obstructions, etc.) So pilots are already used to having to update their information (like charts) periodically. When necessary, the magnetic field information is updated on them. This does mean that the name of a runway might differ from its magnetic orientation over time until it is renamed.
And of course, navigation by non-magnetic means (radio signals, GPS, etc.) are not affected.