This paper presents power allocation schemes to maximize the effective capacity (EC) of a secondary user (SU) communications link using adaptive modulation and coding (AMC) in an underlay cognitive radio Nakagami-m block-fading environment to meet target quality-of-service (QoS) requirements in terms of delay-outage probability and packet error rate constraints. The SU transmission parameters are chosen such that the primary user imposed interference power constraint (IPC) is satisfied. Three different types of IPCs, namely average interference power, peak interference power and interference power outage, are considered. For each IPC, the analytical solutions for choosing the AMC mode and power allocation in each fading block, and the corresponding SU achievable EC under given QoS requirements are derived. Furthermore, we investigate the performance of a hybrid automatic repeat request (ARQ)/AMC and obtain the closed-form packet loss rate expression. Illustrative results show the effects of the IPC, fading duration and fading severeness on the SU achievable EC under given QoS requirements. It is shown that for loose delay-outage requirements, average interference power and interference power outage constraints give higher SU EC than peak interference power constraint. However, for more stringent delay-outage requirements, the SU achievable EC for the three IPC is significantly reduced. The results also indicate that ARQ is helpful to significantly reduce the packet loss rate for loose delay constraint. However, ARQ increases the delay and is not effective for stringent delay-outage requirements.