Commutator of derivatives with torsion

Torsion is always defined by $$ T(X,Y)= nabla_X Y-nabla_y X -[X,Y] $$ and the curvature is defined by the commutator on a vector field as $$ [nabla_X,nabla_Y] Z - nabla_{[X,Y]}Z=R(X,Y)Z. $$ This last equation holds both with and without torsion in the connection.

On a scalar the commutator is given by $$ [nabla_X,nabla_Y]phi - nabla_{[X,Y]}phi= 0 $$ as the scalar does not see curvature. So, using the coordinate basis vectors $X=partial_mu$, $Y=partial_nu$, we have $$ [nabla_{partial_mu},nabla_{partial_nu}]phi=0. $$

It's not safe to write $nabla_{partial_mu}$ as $nabla_mu$ as this gives the impression that you need an extra connection term because of the $nu$ index when $nabla_mu$ acts on $nabla_nu$ etc. If you make this interpretation then then the $nabla_mu$ changes the tensor character of the object it acts on, unlike the usual covariant derivative $nabla_X$ which does not change the tensor character. If you insist of changing the character then $nabla_mu$ and $nabla_nu$ are acting on different spaces depending on their order, and the "commutator" is not really a commutator and its properties are rather ill defined. This is harmless in torsion-free GR but becomes a probelm when you have torsion as it leads to notational ambguities that I suspect you have in your paper.

The torsion definition applied to $X=partial_mu$, $Y=partial_nu$ gives $$ nabla_{partial_mu} partial_nu - nabla_{partial_nu}partial_mu = T^{lambda}(partial_mu,partial_nu) partial_lambda. $$ With $$ T^lambda(partial_mu,partial_nu)= {T^lambda}_{munu} $$ as the coordinate componets of the torsion tensor, and applying the vector field ${T^lambda}_{munu}partial_lambda$ to a scalar $phi$ we get
$$ (nabla_{partial_mu} partial_nu - nabla_{partial_nu}partial_mu) phi = {T^{lambda}}_{munu} partial_lambda phi. $$ which suggests that you opinion that $K$ should be replaced by $T$ is correct.

Also, by the definition of the Christoffel symbols $$ nabla_{partial_mu} partial_nu= {Gamma^lambda}_{numu} partial_lambda $$ we see that the above dequation gives the usual $$ {T^{lambda}}_{munu}= {Gamma^lambda}_{numu}-{Gamma^lambda}_{munu} $$ where I am using MTW's placement of the indices on the $Gamma$'s, so the indices might seem to be backwards compared to some other defs.