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We compute the electric form factors of one-neutron halo nuclei with shallow D-wave states up to next-to-leading order and the E2 transition from the S-wave to the D-wave state up to leading order in Halo effective field theory (Halo EFT). The relevant degrees of freedom are the core and the halo neutron. The EFT expansion is carried out in powers of Rcore/Rhalo, where Rcore and Rhalo denote the length scales of the core and the halo, respectively. We propose a power counting scenario for weakly-bound states in one-neutron Halo EFT and discuss its implications for higher partial waves in terms of universality. The scenario is applied to the 5/2⁺ first excited state and the 1/2⁺ ground state of ¹⁵C. We obtain several universal correlations between electric observables and use data for the E2 transition 5/2⁺ → 1/2⁺ together with ab initio results from the No-Core Shell Model to predict the quadrupole moment.