Speech enhancement using artificial neural networks aims to remove noise from noisy speech signals while preserving the speech content. However, speech enhancement networks often introduce distortions to the speech signal, referred to as artifacts, which can degrade audio quality. In this work, we propose a post-processing neural network designed to mitigate artifacts introduced by speech enhancement models. Inspired by the analogy of making a `Putt' after an `Approach' in golf, we name our model PuttNet. We demonstrate that alternating between a speech enhancement model and the proposed Putt model leads to improved speech quality, as measured by perceptual quality scores (PESQ), objective intelligibility (STOI), and background noise intrusiveness (CBAK) scores. Furthermore, we illustrate with graphical analysis why this alternating Approach outperforms repeated application of either model alone.