This work studies the joint design of cloud and edge processing for the downlink of a fog radio access network (F-RAN). In an F-RAN, cloud processing is carried out by a baseband processing unit (BBU) that is connected to enhanced remote radio heads (eRRHs) by means of fronthaul links. Edge processing is instead enabled by local caching of popular content at the eRRHs. Focusing on the design of the delivery phase for an arbitrary pre-fetching strategy, a novel superposition coding approach is proposed that is based on the hybrid use of the fronthaul links in both hard-transfer and soft-transfer modes. With the former, non-cached files are communicated over the fronthaul links to a subset of eRRHs, while, with the latter, the fronthaul links are used to convey quantized baseband signals as in a cloud RAN (C-RAN). The problem of maximizing the delivery rate is tackled under fronthaul capacity and per-eRRH power constraints. Numerical results are provided to validate the performance of the proposed hybrid delivery scheme for different baseline pre-fetching strategies.