The a posteriori speech presence probability (SPP) is the fundamental component of noise power spectral density (PSD) estimation, which can contribute to speech enhancement and speech recognition systems. Most existing SPP estimators can estimate SPP accurately from the background noise. Nevertheless, numerous challenges persist, including the difficulty of accurately estimating SPP from non-stationary noise with statistics-based methods and the high latency associated with deep learning-based approaches. This paper presents an improved SPP estimation approach based on deep learning to achieve higher SPP estimation accuracy, especially in non-stationary noise conditions. To promote the information extraction performance of the DNN, the global information of the observed signal and the local information of the decoupled frequency bins from the observed signal are connected as hybrid global-local information. The global information is extracted by one encoder. Then, one decoder and two fully connected layers are used to estimate SPP from the information of residual connection. To evaluate the performance of our proposed SPP estimator, the noise PSD estimation and speech enhancement tasks are performed. In contrast to existing minimum mean-square error (MMSE)-based noise PSD estimation approaches, the noise PSD is estimated by the sub-optimal MMSE based on the current frame SPP estimate without smoothing. Directed by the noise PSD estimate, a standard speech enhancement framework, the log spectral amplitude estimator, is employed to extract clean speech from the observed signal. From the experimental results, we can confirm that our proposed SPP estimator can achieve high noise PSD estimation accuracy and speech enhancement performance while requiring low model complexity.