In the paper, we present a cross-layer perspective on data transmission in energy harvesting cognitive radio networks (CRNs). The delay optimal power allocation is studied while taking into account the randomness of harvested energy, data generation, channel state and the grid price. To guarantee primary user (PU)'s transmission, its Signal-Interference-Ratio (SIR) should be no less than a threshold. Each user, including PU as well as secondary user (SU), has energy harvesting devices, and the PU can also purchases the grid power. Each user is rational and selfish to minimize its own the buffer delay. We formulate a stochastic Stackelberg game in a bilevel manner. After decoupling via rewriting the objective and constraints, an equivalent tractable reconstruction is derived. First, we give a distributive algorithm to obtain the Nash equilibrium (NE) of the lower level SUs' noncooperative stochastic game. Thereafter, the stochastic Stackelberg game is discussed under the circumstances that there is no information exchange between PU and SU. Distributed iterative algorithms are designed. Furthermore, a distributive online algorithm is proposed. Finally, simulations are carried out to verify the correctness and demonstrate the effectiveness of proposed algorithms.