The paper considers a spectrum sharing cognitive radio (CR) network coexisting with a primary network under the average interference power constraint. In particular, the secondary user (SU) is assumed to carry delay-sensitive services and thus shall satisfy a given delay quality-of-service (QoS) constraint. The secondary receiver is also assumed to be equipped with multiple antennas to perform maximal ratio combining (MRC) to enhance SU performance. We investigate the effective capacity of the SU with MRC diversity under aforementioned constraints in Nakagami fading environments. Particularly, we derive the optimal power allocation to achieve the maximum effective capacity of the SU, and further derive the effective capacity in closed-form. In addition, we further obtain the closed-form expressions for the effective capacities under three widely used power and rate adaptive transmission schemes, namely, optimal simultaneous power and rate adaptation (opra), truncated channel inversion with fixed rate (tifr) and channel inversion with fixed rate without truncation (cifr). Numerical results supported by simulations are presented to consolidate our studies. The impacts on the effective capacity of various system parameters such as the number of antennas, the average interference power constraint and the delay QoS constraint are investigated in detail. It is shown that MRC diversity can significantly improve the effective capacity of the SU especially for cifr transmission scheme.