Underwater acoustic communications (UAC) features limited bandwidth, distance-dependent attenuation, long and variable propagation delay, and doubly-selective fading. Therefore, the design of reliable and efficient UAC protocols is very challenging. Cooperative relay communications is promising for future UAC as it can improve its reliability and extend the data transmission ranges. In this work, we consider the adaptive relay-aided OFDM UAC (RA-UAC) using the amplifyand- forward (AF) protocol. We first present the results of the optimal power allocation between the source and the relay as well as the power distribution over all subcarriers based on instantaneous channel state information (CSI). However, the adaptive power allocation is very challenging. The fast timevarying UAC channels and the long propagation delay can make the CSI feedback outdated and therefore degrade the performance of the RA-UAC system using the optimal power allocation. To overcome this problem, we implement channel prediction to compensate the channel variation during the CSI signal propagation. The recursive least square (RLS) adaptive filter is utilized to predict the future channel impulse response (CIR) due to its low computational cost and small storage size. The approximate mean square error (MSE) of the RLS filter is derived. Simulation results confirm the necessity of channel prediction through MSE and overall system performance comparison.