Advancements in monaural speech enhancement (SE) techniques have greatly improved the perceptual quality of speech. However, integrating these techniques into automatic speech recognition (ASR) systems has not yielded the expected performance gains, primarily due to the introduction of distortions during the SE process. In this paper, we propose a novel approach called FAT-HuBERT, which leverages distortion-invariant self-supervised learning (SSL) to enhance the robustness of ASR. To address the distortions introduced by the SE frontends, we introduce layer-wise fusion modules that incorporate features extracted from both observed noisy signals and enhanced signals. During training, the SE frontend is randomly selected from a pool of models. We evaluate the performance of FAT-HuBERT on simulated noisy speech generated from LibriSpeech as well as real-world noisy speech from the CHiME-4 1-channel dataset. The experimental results demonstrate a significant relative reduction in word error rate (WER).