In this paper, we investigate resource allocation strategies for a point-to-point wireless communications system with hybrid energy sources consisting of an energy harvester and a conventional energy source. In particular, as an incentive to promote the use of renewable energy, we assume that the renewable energy has a lower cost than the conventional energy. Then, by assuming that the non-causal information of the energy arrivals and the channel power gains are available, we minimize the total energy cost of such a system over $N$ fading slots under a proposed outage constraint together with the energy harvesting constraints. The outage constraint requires a minimum fixed number of slots to be reliably decoded, and thus leads to a mixed-integer programming formulation for the optimization problem. This constraint is useful, for example, if an outer code is used to recover all the data bits. Optimal linear time algorithms are obtained for two extreme cases, i.e., the number of outage slot is $1$ or $N-1$. For the general case, a lower bound based on the linear programming relaxation, and two suboptimal algorithms are proposed. It is shown that the proposed suboptimal algorithms exhibit only a small gap from the lower bound. We then extend the proposed algorithms to the multi-cycle scenario in which the outage constraint is imposed for each cycle separately. Finally, we investigate the resource allocation strategies when only causal information on the energy arrivals and only channel statistics is available. It is shown that the greedy energy allocation is optimal for this scenario.