In the present work, considering critical gravity as a gravity model, an electrically charged topological anti-de Sitter black hole with a matter source characterized by a nonlinear electrodynamics framework is obtained. This configuration is defined by an integration constant, three key structural constants, and a constant that represents the topology of the event horizon. Additionally, based on the Wald formalism, we explore the possibility that this configuration enjoys nontrivial thermodynamic quantities, establishing the corresponding first law of black hole thermodynamics, as well as local stability under thermal and electrical fluctuations. Additionally, via the Gibbs free energy, we note that the topology of the base manifold allows us to compare this charged configuration with respect to the thermal AdS spacetime, allowing us to obtain a Phase Transition. The quasinormal modes and the greybody factor are also calculated by considering the spherical situation. We find that the quasinormal modes exhibit a straightforward change for variations of one of the structural constants.