Proactively pushing content to users has emerged as a promising way of coping with the explosively growing traffic demand of next-generation mobile networks. However, it is still unclear whether content pushing can improve the energy efficiency of hard delay-constrained communications over additive white Gaussian noise (AWGN) channels. With pushing, the energy consumption can be reduced by increasing the available transmission time. But if the user never needs the pushed content, pushing may result in wasted energy. Based on the random content request delay, this paper derives the content request probability threshold that determines whether a content file should be pushed under the hard delay constraint. Moreover, an optimal strategy to allocate transmission power in content pushing and on-demand delivery phases is also proposed. It is shown that the energy efficiency of systems with pushing can be significantly improved as the content request probability increases. Numerical results validate the effectiveness of the proposed power allocation strategy, compared with the on-demand scheme.