Content caching at small-cell base stations (SBSs) is a promising method to mitigate the excessive backhaul load and delay, particularly for on-demand video streaming applications. A cache-enabled heterogeneous cellular network architecture is considered in this paper, where mobile users connect to multiple SBSs during a video downloading session, and the SBSs request files, or fragments of files, from the macro-cell base station (MBS) according to the user requests they receive. A novel coded storage and delivery scheme is introduced to reduce the load on the backhaul link from the MBS to the SBSs. The achievable backhaul delivery rate as well as the number of sub-files required to achieve this rate are studied for the proposed coded delivery scheme, and it is shown that the proposed scheme provides significant reduction in the number of sub-files required, making it more viable for practical applications.