Am I correct in understanding that the GPS satellites can be moved around to suit the requirements of the people who own them i.e. the US military? So if they have operations somewhere they may move the satellites to improve coverage there and potentially leave other areas uncovered?
The GPS satellites are not in geostationary orbit so they have to move about (as seen from the earth) otherwise they'd just fall from the sky. So the military cannot simply move a few satellites from one location to the other, and then keep them there. The only thing they can do is alter their orbits.
There are six intersecting orbits defined, each containing four satellites spread evenly throughout this orbit. The net result of this is that between 6 and 12 satellites are visible from any point on earth simultaneously, assuming that you have a full view of the sky in all directions. But you only need to see four satellites for a reasonably accurate position fix (measured in meters). In that sense, there is no need for the military to move satellites about at random to improve coverage somewhere.
On the other hand, due to the earths uneven field of gravity and all sorts of other cosmic influences, satellites do deviate from their designed orbits. Plus, there are some satellites stacked as 'spares' in higher or lower orbits, ready to take the place of any failed satellite. And if a satellite fails, it has to be moved to a graveyard orbit. For this purpose all satellites will have little booster rockets on board so that the orbit can be adjusted.
GPS satellites regularly broadcast their designed orbit (the "Almanac") and even their current deviation from their defined orbit (that's part of WAAS I think). Your GPS receiver uses this orbit information plus the precise timing signals to determine your position.
So as far as the satellites are concerned, GPS coverage across the globe, including the poles, is fairly consistent and there's no need to adjust the orbits to improve accuracy somewhere.
But there may be atmospheric or terrain issues that cause a degradation in signal. Plus, it's possible for the US military to disable or scramble the GPS signal when the satellites are over a certain area (active warzones for instance) although this will probably limit the Americans ability to wage war too. And it's possible to install local transmitters on the ground which broadcast on the same frequencies as the satellites, leading to local jamming of the signal.