We present the BFV and the BV extension of the Poisson sigma model (PSM) twisted by a closed 3-form H. There exist superfield versions of these functionals such as for the PSM and, more generally, for the AKSZ sigma models. However, in contrast to those theories, they depend on the Euler vector field of the source manifold and contain terms mixing data from the source and the target manifold. Using an auxiliary connection $\nabla $ on the target manifold M, we obtain alternative, purely geometrical expressions without the use of superfields, which are new also for the ordinary PSM and promise adaptations to other Lie algebroid-based gauge theories: The BV functional, in particular, is the sum of the classical action, the Hamiltonian lift of the (only on-shell nilpotent) BRST differential, and a term quadratic in the antifields which is essentially the basic curvature and measures the compatibility of $\nabla $ with the Lie algebroid structure on $T^*M$. We finally construct a $\hbox {Diff}(M)$-equivariant isomorphism between the two BV formulations.