Deformed Gauduchon-Yamabe problems on compact Hermitian manifolds

We study a two-parameter deformation of scalar curvature associated with the Gauduchon connections on compact Hermitian manifolds. A central identity expresses its dependence on the connection parameter through the squared norm of the trace-free part of the torsion of the Chern connection. We establish existence criteria for metrics of constant deformed scalar curvature and analyze the global dependence of the deformed Yamabe invariant on the connection parameter, including when the critical threshold is reached. We also study a prescribed-curvature problem in positive conformal classes. Applications include attainment of the critical threshold for an explicit Hopf example in every complex dimension at least two and on non-locally conformally Kähler Iwasawa products.