Lysosomal dysfunction is linked to several neurodegenerative disorders, in which neurotoxic protein aggregates accumulate in the lysosome and injure lysosomal membranes. To protect cellular homeostasis, injured lysosomes are engulfed and isolated from the cytosol by autophagosomes via a form of selective autophagy, referred to as lysophagy. The mechanisms underlying lysophagy are not well understood. Here, we explore mechanisms regulating lysophagy in HeLa cells and human iPSC-derived neurons (i3Neurons). First, we establish that organelle dynamics within i3Neurons resemble that of primary rat neurons. Then, using this i3Neuron system, we find via immunofluorescence that the selective autophagy adaptor p62/SQSTM1 is highly enriched on damaged lysosomes; in parallel, we find robust p62 recruitment to damaged lysosomes using lysosomal immunoprecipitation and live-cell imaging approaches. We then utilize genetic strategies to show that among the autophagy adaptors, p62 is both necessary and sufficient in lysophagy. A primary function of p62 is to form biomolecular condensates that support local autophagosome biogenesis. We provide the first evidence through biochemical and microscopy-based assays that lysophagy depends on p62 condensates. Importantly, we observe that the properties of p62 condensates are tightly regulated by binding of the small heat shock protein HSP27 to promote lysophagy. Live-cell imaging reveals that HSP27 prevents p62 aggregation on damaged lysosomes. Further, using purified proteins, we determine HSP27 and p62 directly interact, and we identify that this p62-HSP27 interaction is highly regulated in response to lysosomal stress. We observe that this interaction depends on HSP27 phosphorylation. We utilize phosphoproteomics and kinase inhibitor screens to show that a p38 MAPK signaling cascade senses lysosomal injury to phosphorylate HSP27. In sum, we define a novel lysosomal quality control mechanism in which lysosomal injury triggers a p38 MAPK signaling cascade that regulates p62-dependent lysophagy via HSP27. This signaling mechanism senses many cellular stress conditions; therefore, HSP27 may promote many forms of p62-dependent selective autophagy, implicating the p62-HSP27 interaction as a novel, central regulator of selective autophagy.