Hello and welcome to the forums!
The term "impacted wisdom tooth" is actually used to define a wisdom tooth which has grown at an inclined angle towards its neighbouring molar or in any other direction. An impacted wisdom tooth can indeed erupt, although most times only partially. An impacted wisdom tooth, whether partially erupted or totally submerged can indeed be a source of bad breath, because pockets can be created between the tooth and the gum which can become infected. It is also common for infection to occur because the wisdom tooth impacts the neighbouring molar and this creates problems.
There are in fact many problems associated with wisdom teeth, most of them causing bad breath. It is a shame that many dental professionals do not recognise this nowadays (there has been a shift towards trying to keep wisdom teeth in whenever possible rather than pulling them out in the past decade or so…whereas before it was common practice to have them removed at the smallest problem). If a wisdom tooth smells bad, it is a sign that something is not right. Oral hygiene needs to be improved in that area, but if this fails to work, extraction may be the only solution.
Here is a post where I talk about wisdom teeth, infections and how to take care of them.
And yes, a broken tooth is a very common source of bad breath, if there is advanced decay and, particularly, if the root has become infected. Sometimes there is no pain so we think it is not that bad, when in fact it is. Have it checked by your dentist before it is too late!
I hope this helps!
Dolors