In this paper, we investigate the problem of jointly selecting a predefined number of energy-harvesting (EH) sensors and computing the optimal power allocation. The ultimate goal is to minimize the reconstruction distortion at the fusion center. This optimization problem is, unfortunately, non-convex. To circumvent that, we propose two suboptimal strategies: (i) a joint sensor selection and power allocation (JSS-EH) scheme that, we prove, is capable of iteratively finding a stationary solution of the original problem from a sequence of surrogate convex problems; and (ii) a separate sensor selection and power allocation (SS-EH) scheme, on which basis we can identify a sensible sensor selection and analytically find a power allocation policy by solving a convex problem. We also discuss the interplay between the two strategies. Performance in terms of reconstruction distortion, impact of initialization, actual subsets of selected sensors and computed power allocation policies, etc., is assessed by means of computer simulations. To that aim, an EH-agnostic sensor selection strategy, a lower bound on distortion, and an online version of the SS-EH and JSS-EH schemes are derived and used for benchmarking.