We propose a power-and rate-adaptation scheme for cloud radio access networks (C-RANs), where each radio remote head (RRH) is connected to the baseband unit (BBU) pool through optical links. The RRHs jointly support the users by efficiently exploiting the enhanced spatial degrees of freedom. Our proposed scheme aims for maximizing the effective capacity (EC) of the user subject to both per-RRH average-and peak-power constraints, where the EC is defined as the maximum arrival rate that can be supported by the C-RAN under the statistical delay requirement. We first transform the EC maximization problem into an equivalent convex optimization problem. By using the Lagrange dual decomposition method and solving the Karush-Kuhn-Tucker (KKT) equations, the optimal transmission power of each RRH can be obtained in closed-form. Furthermore, an online tracking method is provided for approximating the average power of each RRH for the sake of updating the Lagrange dual variables. For the special case of two RRHs, the expression of the average power of each RRH can be calculated in explicit form. Hence, the Lagrange dual variables can be computed in advance in this special case. Furthermore, we derive the power allocation for two important extreme cases: 1) no delay constraint; 2) extremely stringent delay-requirements. Our simulation results show that the proposed scheme significantly outperforms the conventional algorithm without considering the delay requirements. Furthermore, when appropriately tuning the value of the delay exponent, our proposed algorithm is capable of guaranteeing a delay outage probability below $10^{-9}$ when the maximum tolerable delay is 1 ms. This is suitable for the future ultra-reliable low latency communications (URLLC).