<p>In the paper, a concept of the essential numerical range <i>W</i>_<i>e</i>(<i>T</i>) of a linear relation <i>T</i> in a Hilbert space is given, other various essential numerical ranges <i>W</i>_<i>ei</i>(<i>T</i>), <i>i</i> = 1, 2, 3, 4, are introduced, and relationships among <i>W</i>_<i>e</i>(<i>T</i>) and <i>W</i>_<i>ei</i>(<i>T</i>) are established. These results generalize relevant results obtained by Bögli et al. in [Bögli, S., Marletta, M. and Tretter, C., The essential numerical range for unbounded linear operators, <i>J. Funct. Anal.</i>, <b>279</b>, 2020, 47–12]. Moreover, several fundamental properties of closed relations related to its operator parts are presented. In addition, singular discrete linear Hamiltonian systems including non-symmetric cases are considered, several properties for the associated minimal relations <i>H</i>₀ are derived, and the above results for abstract linear relations are applied to <i>H</i>₀.</p>