In the present paper, we investigate a dependent risk model perturbed by a mixed-exponential jump-diffusion process, in which the claim inter-arrival times and claim sizes are dependent through Farlie-Gumbel-Morgenstern (FGM) copula. The expected discounted penalty (EDP) functions are studied when ruin is caused by a claim or the jump-diffusion process. The Laplace transforms satisfied by the EDP functions are obtained, then we give the corresponding defective renewal equations. The analytical expressions for the EDP functions are derived when the claim sizes follow exponential distributions, and a numerical example for the ruin probabilities are also provided.