As you note $M$ is locally a complete intersection. Take a point and the corresponding local equations near that point, extend those equations to $\mathbb P^m$ so you have a complete intersection subvariety $X\subset \mathbb P^m$ of codimension $r$ such that $X=M\cup N$ for some other subvariety $N\subset \mathbb P^m$ of codimension $r$.

Generally both $X$ and $N$ are singular, but since $X$ is a complete intersection it is Gorenstein and its canonical bundle is computed the same way as in the smooth case. 

If $N$ is a Cartier divisor in $X$, then you have the following relationship:
$$
\omega_M\simeq \omega_X(-N)|_M.
$$
This can be proven by applying the adjunction formula for both $X\subset\mathbb P^m$ and $M\subset\mathbb P^m$. 
Since $M$ and $N$ have no common components, we know that $N|_M$ is effective, so this implies that 
$$\omega_M\subseteq \omega_X|_M.\tag{$\star$}$$

I believe that even if $N$ is not Cartier in $X$ one can say something similar, but it is more complicated. Adding your assumption that $\omega_M$ is a restriction of a line bundle from $\mathbb P^m$ and hence from $X$ should imply that actually $N$ is Cartier in a neighbourhood of $M\cap N$ in $X$, so very likely you have $(\star)$ in that case as well.

Finally a note on the notion of "*subcanonical*". I think by that some people mean a singularity that formally satisfies the definition of a canonical singularity, but with a not necessarily effective divisor as boundary.