We deal with a study of warped product manifold which is also a generalized quasi Einstein manifold. Then, we investigate the relationships between such warped products and certain manifolds that provide some Ricci--Hessian type equations, such as $\operatorname{Ric}_f^m=\lambda g$ for some smooth function $\lambda$, where $\operatorname{Ric}_f^m$ denotes the $m$-Bakery--Emery Ricci tensor. Finally, we obtain some rigidity conditions for such manifolds.