This paper studies the performance of hierarchical modulation-based multimedia transmission in cognitive radio (CR) systems with imperfect channel sensing results under constraints on both transmit and interference power levels. Unequal error protection (UEP) of data transmission using hierarchical quadrature amplitude modulation (HQAM) is considered in which high priority (HP) data is protected more than low priority (LP) data. In this setting, closed-form bit error rate (BER) expressions for HP data and LP data are derived in Nakagami-$m$ fading channels in the presence of sensing errors. Subsequently, the optimal power control that minimizes weighted sum of average BERs of HP bits and LP bits or its upper bound subject to peak/average transmit power and average interference power constraints is derived and a low-complexity power control algorithm is proposed. Power levels are determined in three different scenarios, depending on the availability of perfect channel side information (CSI) of the transmission and interference links, statistical CSI of both links, or perfect CSI of the transmission link and imperfect CSI of the interference link. The impact of imperfect channel sensing decisions on the error rate performance of cognitive transmissions is also evaluated. In addition, tradeoffs between the number of retransmissions, the severity of fading, and peak signal-to-noise ratio (PSNR) quality are analyzed numerically. Moreover, performance comparisons of multimedia transmission with conventional quadrature amplitude modulation (QAM) and HQAM, and the proposed power control strategies are carried out in terms of the received data quality and number of retransmissions.