We develop a projected Nesterov’s proximalgradient (PNPG) scheme for reconstructing sparse signals from compressive Poisson-distributed measurements with the mean signal intensity that follows an affine model with known intercept. The objective function to be minimized is a sum of convex data fidelity (negative log-likelihood (NLL)) and regularization terms. We apply sparse signal regularization where the signal belongs to a nonempty closed convex set within the domain of the NLL and signal sparsity is imposed using total-variation (TV) penalty. We present analytical upper bounds on the regularization tuning constant. The proposed PNPG method employs projected Nesterov’s acceleration step, function restart, and an adaptive stepsize selection scheme that accounts for varying local Lipschitz constant of the NLL.We establish O k2 convergence of the PNPG method with step-size backtracking only and no restart. Numerical examples compare PNPG with the state-of-the-art sparse Poisson-intensity reconstruction algorithm (SPIRAL).