  We study the importance of self-interaction errors in density functional
approximations for various water-ion clusters. We have employed the
Fermi-L\"owdin orbital self-interaction correction (FLOSIC) method in
conjunction with LSDA, PBE, and SCAN to describe binding energies of
hydrogen-bonded water-ion clusters, \textit{i.e.}, water-hydronium,
water-hydroxide, water-halide, as well as non-hydrogen-bonded water-alkali
clusters. In the hydrogen-bonded water-ion clusters, the building blocks are
linked by hydrogen atoms, although the links are much stronger and
longer-ranged than the normal hydrogen bonds between water molecules, because
the monopole on the ion interacts with both permanent and induced dipoles on
the water molecules. We find that self-interaction errors overbind the
hydrogen-bonded water-ion clusters and that FLOSIC reduces the error and brings
the binding energies into closer agreement with higher-level calculations. The
non-hydrogen-bonded water-alkali clusters are not significantly affected by
self-interaction errors. Self-interaction corrected PBE predicts the lowest
mean unsigned error in binding energies ($\leq$ 50 meV/\ce{H2O}) for
hydrogen-bonded water-ion clusters. Self-interaction errors are also largely
dependent on the cluster size, and FLOSIC does not accurately capture the
subtle variation in all clusters, indicating the need for further refinement.
