The focus of this article is studying an optimal control problem for branching diffusion processes. Initially, we introduce the problem in its strong formulation and expand it to include linearly growing drifts. Then, we present a relaxed formulation that provides a suitable characterization based on martingale measures. Considering weak controls, we prove they are equivalent to strong controls in the relaxed setting, and establish the equivalence between the strong and relaxed problem, under a Filippov--type convexity condition. Furthermore, by defining control rules, we can restate the problem as the minimization of a lower semi-continuous function over a compact set, leading to the existence of optimal controls both for the relaxed problem and the strong one. Finally, with a useful embedding technique, we show that the value function solves a system of HJB equations, establishing a verification theorem. We then apply it to a linear-quadratic example and a kinetic one.