I have just started printing lithophanes. I have done a ton of research and found so many different approaches. So i decided to try the lithophane generators online. The best two I found were 'Image to Lithophane' @ 3dp.rocks and one with many user friendly customizations 'www.LithophaneMaker.com'.
After getting the STL file from there, then off to a STL repair site (initially Cura found problems the extensions could not fix) . Since I choose very high definition setting, the STL files were over 200mb and since my computer isn't that good with 3D graphics (it would lock up) , I needed a online repair program.
Freecad, Blender, Tinkercad, Meshmixer, and Meshlab all can fix the STL files but they lock my computer up. So I found a great STL file repair site called 'MakePrintable' (they charge now but not much depending on what you get) and they can take files up to 500mb.
I let them do there thing and pop it back into Cura, and with the extensions that Cura has it finds no holes, and the slicing shows none of the problems that are shown in your picture (use the websites especially of the STL repair sites to get the best result).
I don't have alot of experience but it looks like your layer width, wall thickness, actual resolution of picture and number of walls may need to be looked at. I hope this helps. It surely did with my lithophanes!!!!! Happy printing!!!!!
[gr edit: fixed text color - white on white wasn't visible]