In Lipschitz two-dimensional domains, we study a Brinkman-Darcy-Forchheimer problem on the weighted spaces $\mathbf{H}_0^1(\omega,\Omega) \times L^2(\omega,\Omega)/\mathbb{R}$, where $\omega$ belongs to the Muckenhoupt class $A_2$. Under a suitable smallness assumption, we establish the existence and uniqueness of a solution. We propose a finite element scheme and obtain a quasi-best approximation result in energy norm \`a la C\'ea under the assumption that $\Omega$ is convex. We also devise an a posteriori error estimator and investigate its reliability and efficiency properties. Finally, we design a simple adaptive strategy that yields optimal experimental rates of convergence for the numerical examples that we perform.

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