"To cover the surface" means to completely overlay or spread something over the entire outer area or topmost layer of an object or space. Full definition
Keep going until you've completely covered the surface of the ice with your paint. (unfolded.com)
Pour into a clean bowl and immediately cover the surface with plastic wrap to prevent a crust from forming. (joyofbaking.com)
The light conditions (from overhead or streaming through the gallery's long window) affect the viewer's already partial perspective by covering the surfaces in uneven shadows. (frieze.com)