In this work, we conducted research on deformable object manipulation by robots based on demonstration-enhanced reinforcement learning (RL). To improve the learning efficiency of RL, we enhanced the utilization of demonstration data from multiple aspects and proposed the HGCR-DDPG algorithm. It uses a novel high-dimensional fuzzy approach for grasping-point selection, a refined behavior-cloning method to enhance data-driven learning in Rainbow-DDPG, and a sequential policy-learning strategy. Compared to the baseline algorithm (Rainbow-DDPG), our proposed HGCR-DDPG achieved 2.01 times the global average reward and reduced the global average standard deviation to 45% of that of the baseline algorithm. To reduce the human labor cost of demonstration collection, we proposed a low-cost demonstration collection method based on Nonlinear Model Predictive Control (NMPC). Simulation experiment results show that demonstrations collected through NMPC can be used to train HGCR-DDPG, achieving comparable results to those obtained with human demonstrations. To validate the feasibility of our proposed methods in real-world environments, we conducted physical experiments involving deformable object manipulation. We manipulated fabric to perform three tasks: diagonal folding, central axis folding, and flattening. The experimental results demonstrate that our proposed method achieved success rates of 83.3%, 80%, and 100% for these three tasks, respectively, validating the effectiveness of our approach. Compared to current large-model approaches for robot manipulation, the proposed algorithm is lightweight, requires fewer computational resources, and offers task-specific customization and efficient adaptability for specific tasks.