Writing your question in a more familiar language for me: Given

$D_X Y=X^\mu\partial_\mu Y^\nu+A^{\nu\mu\rho}X_\mu Y_\rho$

$D'_X Y=X^\mu\partial_\mu Y^\nu+B^{\nu\mu\rho}X_\mu Y_\rho$

$A^{\nu\mu\rho}=A^{\nu\rho\mu}$

$B^{\nu\mu\rho}=B^{\nu\rho\mu}$

$\partial^\mu g_{\nu\rho}=A^{\nu\mu\rho}+B^{\nu\mu\rho}$

$\partial^\mu w_{\nu\rho}=A^{\tau\mu\nu} w_\tau\,^\rho+B^{\tau\mu\rho} w^\nu\,_\tau$

Can we reflect the variables $A$ and $B$? Because if we can, calculating the curvature is almost trivial. At least we know that $A^\mu\,_{\nu\rho}+B^\mu\,_{\nu\rho}=2\Gamma^\mu\,_{\nu\rho}$ (the Christoffel Symbols from the levi-civita conection). I'm not sure how to solve the last equation, now I don't have enough time to do that, sorry.