In this article we study, via analytical methods, 1/Q non-perturbative power corrections to event shape mean values, addressing in particular the question of their interplay with soft perturbative emissions. Specifically we point out that energy-ordered soft perturbative emissions that precede a non-perturbative emission, give rise to terms of the form \( \frac{1}{Q}{\left({\alpha}_s\ln \frac{Q}{\Lambda}\right)}^n \) . While such terms are formally higher order in the strong coupling, their form suggests that they can numerically compete with the leading 1/Q term while also modifying the Q dependence of the result. The resummation of such power-suppressed but logarithmically enhanced terms lends an anomalous dimension to the leading 1/Q power correction. In order to argue for the presence of such an anomalous dimension, we formulate a method to compute the first order in αs correction for the mean values of the thrust 1 – T and C-parameter observables. We comment on our findings in light of the standard picture of universality of 1/Q power corrections for event shape variables and implications for phenomenology.