In the present contribution, we propose a novel opportunistic ambient backscatter communication (ABC) framework for radio frequency (RF)-powered cognitive radio (CR) networks. This framework considers opportunistic spectrum sensing integrated with ABC and harvest-then-transmit (HTT) operation strategies. Novel analytic expressions are derived for the average throughput, the average energy consumption and the energy efficiency in the considered set up. These expressions are represented in closed-form and have a tractable algebraic representation which renders them convenient to handle both analytically and numerically. In addition, we formulate an optimization problem to maximize the energy efficiency of the CR system operating in mixed ABC $-$ and HTT $-$ modes, for a given set of constraints including primary interference and imperfect spectrum sensing constraints. Capitalizing on this, we determine the optimal set of parameters which in turn comprise the optimal detection threshold, the optimal degree of trade-off between the CR system operating in the ABC $-$ and HTT $-$ modes and the optimal data transmission time. Extensive results from respective computer simulations are also presented for corroborating the corresponding analytic results and to demonstrate the performance gain of the proposed model in terms of energy efficiency.