We introduce the notion of an $ls$-$\pi$-Ponomarev-system to give necessary and sufficient conditions for $f:(M,M_0)\to X$ to be a strong $wc$-mapping ($wc$-mapping, $wk$-mapping) where $M$ is a locally separable metric space. Then, we systematically get characterizations of weakly continuous strong $wc$-images ($wc$-images, $wk$-images) of locally separable metric spaces by means of certain networks. Also, we give counterexamples to sharpen some results on images of locally separable metric spaces in the literature.