Currently, the robustness of precast reinforced concrete frames is attracting wide attention. However, avoiding “strong beams and weak columns” during strengthening against progressive collapse is a key problem. To discuss this problem, an experimental study on two 1/2-scale precast frame subassemblages under a pushdown loading regime was carried out in this paper. One specimen was strengthened with carbon fibre-reinforced polymer (CFRP) sheets on the beam sides. The middle parts of the CFRP sheets were anchored with hybrid fibre-reinforced polymer (HFRP) anchors. Another specimen was not strengthened. The failure mechanisms, failure modes, and strengthening effect are discussed. The strengthening effect is very obvious in the early catenary action stage. No shearing failure develops on HFRP anchors, which proves that the anchoring method is effective. Based on the experimental results, analytical models and preventive strengthening design and construction measures to mitigate progressive collapse of the precast RC frame are proposed.