Strengthening existing masonry arch structures with externally bonded composite systems has become a common technique to enhance structural performance while allowing continued use of the infrastructure during retrofitting. Among these systems, textile-reinforced mortar (TRM) solutions applied to the intrados of arches have shown promising results. The effectiveness of mortar-based materials, such as fabric reinforced cementitious matrix (FRCM), TRM, and steel reinforced grout (SRG), relies heavily on the interaction between the matrix, the masonry substrate, and the embedded fibres. However, when applied to concave masonry surfaces, the curvature introduces additional normal stresses that may hinder stress transmission and bond performance. This study investigates how substrate curvature and the use of spike anchors influence the failure mechanisms in FRCM strengthened masonry. Single shear-lap tests were conducted on curved masonry specimens to analyse load transfer behavior and identify failure modes. The results reveal that curvature significantly alters typical failure mechanisms observed on flat substrates, even leading to the emergence of subcategories within standard failure modes. Moreover, the incorporation of spike anchors was found to locally modify failure progression, improving overall bond performance. The paper presents a detailed classification of observed failure modes, supported by histograms that quantify the effect of curvature and the spike anchors. These findings provide valuable insight into the design of strengthening systems for curved masonry elements and highlight the importance of tailored anchorage strategies.