If you use the online address search feature, Lutherstadt Wittenberg is pinpointed correctly. I checked with Openstreetmap.org and can’t find any such issue there. So the issue seems with some other database used by PE.
If you do a search for Dessau, it brings you correctly to that city.
Your right, we are using data from geonames.org to build our Destinations DB and excludes islands and small towns. Also as to not require a lot of extra data for exact administrative boundaries for cities, etc PocketEarth is not always able to give the correct town name, sometimes picking a bigger city nearby. Using geonames data also has other downsides such as not correctly localizing place names to the user’s language.
We are working to replace our Destinations DB with information straight from OSM which will correct all of these issues!