We analyze energetics of a non-Gaussian process described by a stochastic differential equation of the Langevin type. The process represents a paradigmatic model of a nonequilibrium system subject to thermal fluctuations and additional external noise, with both sources of perturbations considered as additive and statistically independent forcings. We define thermodynamic quantities for trajectories of the process and analyze contributions to mechanical work and heat. As a working example we consider a particle subjected to a drag force and two statistically independent Lévy white noises with stability indices =2 and =1. The fluctuations of dissipated energy (heat) and distribution of work performed by the force acting on the system are addressed by examining contributions of Cauchy fluctuations (\alpha = 1) to either bath or external force acting on the system.