We introduce the notion of an $ls$-$\sigma$-finite Ponomarev-system $(f, M, X, \{\mathcal P_{\lambda,n}\})$ to give a consistent method to construct an $s$-mapping (msss-mapping, mssc-mapping, cs-mapping) f with covering-properties from a locally separable metric space $M$ onto a space $X$. As applications, we systematically get characterizations of $s$-images (msss-images, mssc-images, cs-images) with covering-properties of locally separable metric spaces.