In this article, we describe the endomorphism ring of a finitely generated progenerator module of a weighted Leavitt path algebra \(L_{\textsf{k}}(E, w)\) of a finite vertex-weighted graph (E, w). Contrary to the case of Leavitt path algebras, we show that a (full) corner of a weighted Leavitt path algebra is, in general, not isomorphic to a weighted Leavitt path algebra. However, using the above result, we show that for every full idempotent \(\epsilon \) in \(L_{\textsf{k}}(E, w)\) , there exists a positive integer n such that \({\mathbb {M}}_n(\epsilon L_{\textsf{k}}(E, w) \epsilon )\) is isomorphic to the weighted Leavitt path algebra of a weighted graph explicitly constructed from (E, w). We then completely describe unital algebras being Morita equivalent to weighted Leavitt path algebras of vertex-weighted graphs. In particular, we characterize unital algebras being Morita equivalent to sandpile algebras.
