In a Fog Radio Access Network (Fog-RAN), edge caching is combined with cloud-aided transmission in order to compensate for the limited hit probability of the caches at the base stations (BSs). Unlike the typical wired scenarios studied in the networking literature in which entire files are typically cached, recent research has suggested that fractional caching at the BSs of a wireless system can be beneficial. This paper investigates the benefits of fractional caching in a scenario with a cloud processor connected via a wireless fronthaul link to a BS, which serves a number of mobile users on a wireless downlink channel using orthogonal spectral resources. The fronthaul and downlink channels occupy orthogonal frequency bands. The end-to-end delivery latency for given requests of the users depends on the HARQ processes run on the two links to counteract fading-induced outages. An analytical framework based on theory of Markov chains with rewards is provided that enables the optimization of fractional edge caching at the BSs. Numerical results demonstrate meaningful advantages for fractional caching due to the interplay between caching and HARQ transmission. The gains are observed in the typical case in which the performance is limited by the wireless downlink channel and the file popularity distribution is not too skewed.