In addition to triggering natural earthquakes, (as pointed out by @hichris123) Mines can also be sources of artificial seismic activities due to mine collapses and/or explosives.
Miners will generally try their best to keep their mine shaft stable, but their methods aren't 100% accurate. Sometimes mining in an unstable area or with the lack of support to a mine will cause the shaft to collapse, killing anyone still in it. When this happens, there can be a chance that an earthquake could be felt from a nearby town or city.
Are earthquakes more common in mining regions than they would otherwise be?
While there can be earthquakes in mining regions due to other reasons, yes, they are. When a mine collapses, the releasing of the rocks can be so abrupt that kinetic energy is discharged in the form of a low-magnitude earthquake. This is also known as a form of induced seismicity, and earthquakes resulting from induced seismicity generally aren't that serious.
When you google 'earthquakes due to mining', the following blurb will appear:
The six miners and three rescuers that were killed in the magnitude-3.9 earthquake due to the collapse of the Crandall Coal mine are now memorialized at the site. ... The abrupt release of elastic strain in the rocks during a mine collapse discharges energy in the form of seismic waves.
So this is a good example of induced seismicity.
e.g. is the frequency of earthquakes in those regions different when mining is occurring than when it is not?
Since you mentioned mining regions, it will be assumed that a collapsing mine would be more of a common occurrence in a mining region than anywhere else. So yes, the difference would be that there are more seismic waves creating earthquakes in a mining region compared to you're average city, because less mining is occurring in a city that in a mountain. But when no mining is in progress an a mining region, there will be less human-induced earthquakes.