The full physics potential of the next-generation Deep Underground Neutrino Experiment (DUNE) is still being explored. In particular, there have been some recent studies on the possibility of improving DUNEs neutrino energy reconstruction. The main motivation is that a better determination of the neutrino energy in an event-by-event basis will translate into an improved measurement of the Dirac $CP$ phase and other neutrino oscillation parameters. To further motivate studies and improvements on the neutrino energy reconstruction, we evaluate the impact of energy resolution at DUNE on an illustrative new physics scenario, viz. non-standard interactions (NSI) of neutrinos with matter. We show that a better energy resolution in comparison to the ones given in the DUNE conceptual and technical design reports may significantly enhance the experimental sensitivity to NSI, particularly when degeneracies are present. While a better reconstruction of the first oscillation peak helps disentangling standard $CP$ effects from those coming from NSIs, we find that the second oscillation peak also plays a nontrivial role in improving DUNEs sensitivity.
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