Flattening a connection on a Kähler manifold

Question

Say $M$ is a closed Kähler manifold and $(V, \nabla)$ is a (say) constant Hermitian bundle on $V$ with (say) trivial flat connection. Now $M$ Kähler gives several distinguished classes of closed one-forms in $\Omega^1(M, \mathrm{End}(V))$ (harmonic, holomorphic, and variations on these). I'm curious whether there is a special class of one-forms for which the connection $\nabla + \hbar\eta$ (which is flat to second order) can be canonically deformed to a flat connection $\nabla + \hbar\eta + O(\hbar^2)$. Is there some condition that guarantees this? Is there a context where the deformation theory becomes easily tractable? (I am assuming that $M$ is Kähler here because I know Hodge theory makes deformation theory works better on Kähler manifolds - if there is an answer in the more general case where $M$ is Riemannian and $\eta$ is harmonic, I'm also curious about that.)