This work presents a single-channel speech enhancement (SE) framework based on the super-Gaussian extension of the joint maximum a posteriori (SGJMAP) estimation rule. The developed SE algorithm is an open-source research smartphone-based application for hearing improvement studies. In this algorithm, the SGJMAP-based estimation for noisy speech mixture is smoothed along the frequency axis by a Mel filter-bank, resulting in a Mel-warped frequency-domain SGJMAP estimation. The impulse response of this Mel-warped estimation is obtained by applying a Mel-warped inverse discrete cosine transform (Mel-IDCT). This helps in filtering out the background noise and enhancing the speech signal. The proposed application is implemented on an iPhone (Apple, Cupertino, CA) to operate in real time and tested with normal-hearing (NH) and hearing-impaired (HI) listeners with different types of hearing aids through wireless connectivity. The objective speech quality and intelligibility test results are used to compare the performance of the proposed algorithm to existing conventional single-channel SE methods. Additionally, test results from NH and HI listeners show substantial improvement in speech recognition with the developed method in simulated real-world noisy conditions at different signal-to-noise ratio levels.