Let $H$ be a weak Hopf $\pi$-coalgebra and let $A$ be a weak right $\pi$-$H$-comodule algebra with a total integral $\varphi$. This paper is mainly devoted to a Maschke-type theorem for $(H,A)$-weak relative Hopf $\pi$-modules. As an application, we obtain a characterization for an $(H,A)$-weak Hopf $\pi$-module to be projective as an $A$-module.