A connected graph is said to be supereulerian if it containes a spanning Eulerian subgraph. The interest in these graphs comes from the fact that such a graph has a Hamiltonian line graph. A graph \(G\) is called essentially \(k\)-edge connected if \(G\) does not have an essential edge cut \(X\) with \(|X| < k.\) The main results of this paper characterize the structure of essentially 2-edge-connected graphs with nonsupereulerian line graphs. For example, the authors present the following results. Theorem 1. Let \(G\) be an essentially 2-connected graph. Then either \(L(G)\) is supereulerian or \(G\) is contractible to a graph that contains one of four small subgraphs described by the authors as an induced subgraph. Corollary 2. If \(G\) is essentially 2-edge-connected with at most five vertices of degree two, then \(L(G)\) is supereulerian. The authors also correct a flaw in a charaterization of graphs with supereulerian line graphs given by \textit{Y. Huang} et al. [``A characterization of graphs with supereulerian line graphs'', Int. J. Comput. Math. Comput. Syst. Theory 5, 1--14 (2020; \url{doi:10.1080/23799927.2019.1708465})].