Accurate and complete genome replication is a fundamental cellular process for the proper transfer of genetic material to cell progenies, normal cell growth, and genome stability. However, a plethora of extrinsic and intrinsic factors challenge individual DNA replication forks and cause replication stress (RS), a hallmark of cancer. When challenged by RS, cells deploy an extensive range of mechanisms to safeguard replicating genomes and limit the burden of DNA damage. Prominent among those is homologous recombination (HR). Although fundamental to cell division, evidence suggests that cancer cells exploit and manipulate these RS responses to fuel their evolution and gain resistance to therapeutic interventions. In this review, we focused on recent insights into HR-mediated protection of stress-induced DNA replication intermediates, particularly the repair and protection of daughter strand gaps (DSGs) that arise from discontinuous replication across a damaged DNA template. Besides mechanistic underpinnings of this process, which markedly differ depending on the extent and duration of RS, we highlight the pathophysiological scenarios where DSG repair is naturally silenced. Finally, we discuss how such pathophysiological events fuel rampant mutagenesis, promoting cancer evolution, but also manifest in adaptative responses that can be targeted for cancer therapy.