We consider a cache-enabled heterogeneous cellular network, where mobile users (MUs) connect to multiple cache-enabled small-cell base stations (SBSs) during a video downloading session. SBSs can deliver these requests using their local cache contents as well as by downloading them from a macro-cell base station (MBS), which has access to the file library. We introduce a novel mobility-aware content storage and delivery scheme, which jointly exploits coded storage at the SBSs and coded delivery from the MBS to reduce the backhaul load from the MBS to the SBSs. We show that the proposed scheme provides a significant reduction both in the backhaul load when the cache capacity is sufficiently large, and in the number of sub-files required. Overall, for practical scenarios, in which the number of sub-files that can be created is limited either by the size of the files, or by the protocol overhead, the proposed coded caching and delivery scheme decidedly outperforms state-of-the-art alternatives. Finally, we show that the benefits of the proposed scheme also extends to scenarios with non-uniform file popularities and arbitrary mobility patterns.