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, so
$$
\omega_X\simeq \mathscr O_{\mathbb P^m}(d-m-1)|_X\tag{$\star$}
$$
where $d$ is the sum of the degree of equations defining $X$.
Claim $\quad$
$\omega_M\subseteq \mathscr O_{\mathbb P^m}(d-m-1)|_M$ where $d$ is the minimum of the sum of degrees of local defining equations for $M$.
Notation: For a subvariety $Z\subseteq \mathbb P^m$, denote the ideal sheaf of $Z$ by $\mathscr I_Z$.
Remark: Note that this claim does not require that $\omega_M$ is a line bundle restricted from $\mathbb P^m$. See also the corollary.
Proof:
Since $M\subseteq X$, we have $\mathscr I_X\subseteq \mathscr I_M$ and hence we get a natural morphism
$$
\iota: (\mathscr I_X/\mathscr I_X^2)|_M \to \mathscr I_M/\mathscr I_M^2.
$$
As $X$ is a complete intersection, $\mathscr I_X/\mathscr I_X^2$ is locally free, and hence in particular it is torsion-free. Furthermore, $\iota$ is an isomorphism on $M\setminus N$ which is an open dense subset of $M$.
It follows that $\ker\iota$ is a torsion subsheaf of $\mathscr I_X/\mathscr I_X^2$, so $\ker\iota=0$ and thus $\iota$ is an injection on all of $M$.
As $M$ is smooth,
$$
\det\mathscr N_{M/\mathbb P^m}=(\det \mathscr I_M/\mathscr I_M^2)^*
$$
is a line bundle, so taking duals and determinants we get that
$$
\det\mathscr N_{M/\mathbb P^m}\subseteq \det\mathscr N_{X/\mathbb P^m}|_M.
$$
By applying the adjunction formula for both $X\subset\mathbb P^m$ and $M\subset\mathbb P^m$,
we get hat
$$\omega_M\subseteq \omega_X|_M.$$
The Claim follows by $(\star)$. $\quad\square$
Corollary $\quad$ If $\omega_M$ is a line bundle restricted from $\mathbb P^m$, then
$\omega_M\simeq \mathscr O_{\mathbb P^m}(q)$ for some $q\leq d-m-1$ where $d$ is as above.
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.
