Reducing noise interference is crucial for automatic speech recognition (ASR) in a real-world scenario. However, most single-channel speech enhancement (SE) generates "processing artifacts" that negatively affect ASR performance. Hence, in this study, we suggest a Noise- and Artifacts-aware loss function, NAaLoss, to ameliorate the influence of artifacts from a novel perspective. NAaLoss considers the loss of estimation, de-artifact, and noise ignorance, enabling the learned SE to individually model speech, artifacts, and noise. We examine two SE models (simple/advanced) learned with NAaLoss under various input scenarios (clean/noisy) using two configurations of the ASR system (with/without noise robustness). Experiments reveal that NAaLoss significantly improves the ASR performance of most setups while preserving the quality of SE toward perception and intelligibility. Furthermore, we visualize artifacts through waveforms and spectrograms, and explain their impact on ASR.