To support delay-bounded multimedia services over 5G mobile wireless networks, the statistical quality-of-service (QoS) technique has been designed to jointly guarantee statistically delay-bounded video transmissions over different time-varying wireless channels, simultaneously. On the other hand, energy harvesting (EH) wireless channels/networks have received a great deal of research attention recently to address the issue of terminals that need to refresh their energy supplies remotely. An EH device is considered as a rechargeable battery which stores the incoming energy from various external sources. As one of the 5G promising technologies, the study of EH systems has brought many new challenges, such as how to characterize the wireless channels for EH systems using finite blocklength coding (FBC). In addition, given blocklength n, error probability E, and E-effective capacity in the finite blocklength regime, the achievable data transmission rate under statistical delay-bounded and error-rate bounded QoS requirements for EH based systems still remains as a challenging and open problem. To overcome the aforementioned problems, we propose FBC based cross-layer design for EH systems in supporting statistical delay-bounded and error-rate bounded QoS requirements over 5G mobile wireless networks. In particular, we establish and analyze FBC based EH system models. Given statistical delay-bounded and error-rate bounded QoS constraints, we derive an approximate lower bound on the data transmission rate in terms of the effective capacity for our proposed EH system in the finite blocklength regime. Also conducted is a set of simulations that validate and evaluate our proposed FBC based EH system under statistical delay-bounded and error-rate bounded QoS constraints.