Important Notes

When using a Clay modifier, material has to be assigned via the modifier’s Material input. Assigning the material to your object’s empty material slot won’t work, do it only if you want to access the material’s parameters.
Right-click on your object and select Shade Smooth. This will fix “hard edges” and “split faces”.
To make the textures “stick” to your object:

Drag-n-drop the ‘Sticky Textures’ modifier on to the object.
Make sure this modifier is above any ‘Armature’ or deformation modifier in the Modifiers tab.
In the Clay Doh material settings, under “Texture Mapping”, turn “Sticky Textures” to 1.

When using ‘Curve > Clay’ modifier, select your curve and under the Curve Data tab, set the ‘Fill Mode’ to None. When ‘Fill Mode’ is set to anything else, the Curve turns into a Mesh.
When using ‘Mesh > Clay’ modifier, if there are any issues/ artifacts, please select all faces in Edit Mode, press Alt+N and select ‘Recalculate Outside’. This will fix any issues with face normals.