This letter establishes a unified analytical framework to study the asymptotic average mutual information (AMI) of mixture gamma (MG) distributed fading channels driven by finite input signals in the high signal-to-noise ratio (SNR) regime. It is found that the AMI converges to some constant as the average SNR increases and its rate of convergence (ROC) is determined by the coding gain and diversity order. Moreover, the derived results are used to investigate the asymptotic optimal power allocation policy of a bank of parallel fading channels having finite inputs. It is suggested that in the high SNR region, the sub-channel with a lower coding gain or diversity order should be allocated with more power. Finally, numerical results are provided to collaborate the theoretical analyses.