In recent decades, many studies have suggested that phase information is crucial for speech enhancement (SE), and time-domain single-channel speech enhancement techniques have shown promise in noise suppression and robust automatic speech recognition (ASR). This paper presents a continuation of the above lines of research and explores two effective SE methods that consider phase information in time domain and frequency domain of speech signals, respectively. Going one step further, we put forward a novel cross-domain speech enhancement model and a bi-projection fusion (BPF) mechanism for noise-robust ASR. To evaluate the effectiveness of our proposed method, we conduct an extensive set of experiments on the publicly-available Aishell-1 Mandarin benchmark speech corpus. The evaluation results confirm the superiority of our proposed method in relation to a few current top-of-the-line time-domain and frequency-domain SE methods in both enhancement and ASR evaluation metrics for the test set of scenarios contaminated with seen and unseen noise, respectively.