For a given Riemmanian connection defined on a smooth manifold $M$, we denote its covariant derivative by $D_V$ where $V\in \mathcal{x}(M)$, the smooth vector fields on this manifold.
Then is it possible to give an explicit coordinate-free expression of $D_{V}[X,Y]$ where $X,Y\in \mathcal{x}(M)$?
I want to know if there is some explicit coordinate-free expression involving, again, Lie brackets and Riemannian metric and derivatives and some other 'general-type symbols'. That is to say, I want to know if there is any related result concerning this $D_{V}[X,Y]$.
For example I would like something like $D_{V}[X,Y]=[D_{V} X,Y]+[X, D_V{Y}]$ (Of course this 'Leibniz rule' is NOT correct if you check it using a spherical coordinate)
I am assuming that by "for a given Riemannian connection on $M$," you mean "given the Riemannian connection on $M$" (i.e., the Levi-Civita connection) . . .
Requiring that the connection be torsion free (which is a requirement of the Levi-Civita connection) is equivalent to requiring that $$ D_{X}Y - D_{Y}X = \left[ X , Y\right] $$ for all vector fields $X, Y$ on $M$.
It follows that $$ D_{V} \left[ X , Y\right] = D_{V}\left(D_{X}Y - D_{Y}X\right) = D_{V}D_{X} Y - D_{V} D_{Y} X, $$ which is (I think) the best that you can do.