Optimization problems constrained by partial differential equations (PDEs) are ubiquitous in science and engineering, arising as optimal control, design and inverse problems. These problems are notoriously challenging to solve numerically. For example, simply evaluating the objective function requires solving a large-scale system of equations resulting from the discretized PDE. This exorbitant cost necessitates the use of rapidly converging optimization routines to reduce the number of evaluations of the objective function and its derivatives. Unfortunately, this expense is exacerbated when the objective function involves nonsmooth terms such sparsifying regularizers or risk measures for stochastic problems. Traditional nonsmooth optimization methods converge (sub)linearly, often requiring many iterations to achieve marginal accuracy. In this talk, we introduce a proximal trust-region Newton method for minimizing a specific class of structured nonsmooth objective functions in Hilbert space. Our method is unique in that it permits and systematically controls inexact objective function and derivative evaluations. We prove global convergence of our method and establish that it converges superlinearly, even quadratically, for specific objective function types. We demonstrate the efficiency of our algorithm on various risk-averse PDE-constrained optimization examples as well as the simulation of a viscoelastic flow.