<p>The present research explored the complex dynamics of an interacting species system influenced by significant ecological phenomena, namely, prey refuge depending upon both the species and weak Allee effect on the predator species. The proposed system’s dynamics comprise bubbling and hydra effects encompassing more than usual predator–prey interaction, subject to significant system parameters. Firstly, ecologically meaningful equilibria have been mathematically analyzed by thoroughly examining their stability, instability, and potential bifurcation scenarios, which demonstrates complex dynamic scenarios related to bi-stability, tri-stability, and various codimension 1 and 2 bifurcations, including transcritical, saddle-node, Hopf, cusp, and Bogdanov–Takens bifurcations. It has been revealed through extensive numerical instances that the Allee and refuge effects introduce complexity into the model. Secondly, quantitative frameworks indicate the emergence of spatiotemporal patterns both within and outside Turing space. Furthermore, the evolution of diffusion-driven spatiotemporal pattern is depicted on a two-dimensional plane, showcasing Turing patterns, such as spot pattern, stripe pattern, the mixture pattern of spots and stripes, the mixture pattern of short stripes and spirals and the octagonal pattern, induced by pure Turing instability; and non-Turing patterns, such as black-eye, spiral wave, combinations of stripes and cold spots pattern, induced by Turing-Hopf instability and Turing-saddle-node instability respectively. The present analysis suggests that bubble and hydra effects provide new insights into the spatiotemporal dynamics of the predator–prey model, thereby deepening the understanding of its dynamic behaviour.</p>