In this paper, we study the design of a hybrid precoder, consisting of an analog and a digital precoder, for the delivery phase of downlink cache-enabled millimeter wave (mmWave) radio access networks (CeMm-RANs). In CeMm-RANs, enhanced remote radio heads (eRRHs), which are equipped with local cache and baseband signal processing capabilities in addition to the basic functionalities of conventional RRHs, are connected to the baseband processing unit via fronthaul links. Two different fronthaul information transfer strategies are considered, namely, hard fronthaul information transfer, where hard information of uncached requested files is transmitted via the fronthaul links to a subset of eRRHs, and soft fronthaul information transfer, where the fronthaul links are used to transmit quantized baseband signals of uncached requested files. The hybrid precoder is optimized for maximization of the minimum user rate under a fronthaul capacity constraint, an eRRH transmit power constraint, and a constant-modulus constraint on the analog precoder. The resulting optimization problem is non-convex, and hence the global optimal solution is difficult to obtain. Therefore, convex approximation methods are employed to tackle the non-convexity of the achievable user rate, the fronthaul capacity constraint, and the constant modulus constraint on the analog precoder. Then, an effective algorithm with provable convergence is developed to solve the approximated optimization problem. Simulation results are provided to evaluate the performance of the proposed algorithms, where fully digital precoding is used as benchmark. The results reveal that except for the case of a large fronthaul link capacity, soft fronthaul information transfer is preferable for CeMm-RANs.