Edit: This is wrong. Sorry.
The minimal generators of the module $\mathrm {coker}\ M$ are the column vectors $(v,x)^T$ and $(y,z)^T$. They generate a two-dimensional vector space of all the minimal generators of the module. This is just $X/mX$, where $X = \mathrm{coker}\ M$ and $m=(x,y,z,v)$. If there is going to be a direct summand isomorphic to $k$, there must be a minimal generator which is annihilated by the maximal ideal. But one can write down a generic minimal generator $(av+by, ax+bz)^T$ and the 8 $k$-linear equations saying that it is annihilated by $x,y,z$ and $v$. Two of them are $v(av+by)=0$ and $z(ax+bz)=0$. The relations in the ring imply $bvy=0=avx$. Since $vy$ and $vx$ are nonzero in $R$, this means $a=0=b$, and so there is no such direct summand.
Would you tell us why you thought there should be such a direct summand?