I will work over a field of characteristic $0$ so that reductive algebraic groups are linearly reductive; presumably there is a way to eliminate this hypothesis. In this case, for an integer $n>1$ and for a divisor $m$ such that $n>m>1$, there **does not** exist $f$ as above. The point is to consider the critical locus of the determinant. The computation below proves that "cohomology with supports" has cohomological dimension equal to $2(n-1)$ for the pair of the quasi-affine scheme $\textbf{Mat}_{n\times n} \setminus \text{Crit}(\Delta_{n\times n})$ and its closed subset $\text{Zero}(\Delta_{n\times n})\setminus \text{Crit}(\Delta_{n\times n})$. Since pushforward under affine morphisms preserve cohomology of quasi-coherent sheaves, that leads to a contradiction.

Denote the $m\times m$ determinant polynomial by $$\Delta_{m\times m}:\textbf{Mat}_{m\times m} \to \mathbb{A}^1.$$ For integers $m$ and $n$ such that $m$ divides $n$, you ask whether there exists a homogeneous polynomial morphism $$f:\textbf{Mat}_{n\times n}\to \textbf{Mat}_{m\times m}$$ of degree $n/m$ such that $\Delta_{n\times n}$ equals $\Delta_{m\times m}\circ f$. Of course that is true if $m$ equals $1$: just define $f$ to be $\text{Delta}_{n\times n}$. Similarly, this is true if $m$ equals $n$: just define $f$ to be the identity. Thus, assume that $2\leq m < n$; this manifests below through the fact that the critical locus of $\Delta_{m\times m}$ is nonempty. By way of contradiction, assume that there exists $f$ with $\Delta_{n\times n}$ equal to $\Delta_{m\times m}\circ f$.

**Lemma 1.** The inverse image under $f$ of $\text{Zero}(\Delta_{m\times m})$ equals $\text{Zero}(\Delta_{n\times n})$. In other words, the inverse image under $f$ of the locus of matrices with nullity $\geq 1$ equals the locus of matrices with nullity $\geq 1$.

**Proof.** This is immediate. **QED**

**Lemma 2.** The inverse image under $f$ of the critical locus of $\Delta_{m\times m}$ equals the critical locus of $\Delta_{n\times n}$. In other words, the inverse image under $f$ of the locus of matrices with nullity $\geq 2$ equals the locus of matrices with nullity $\geq 2$.

**Proof.**
By the Chain Rule, $$d_A\Delta_{n\times n} = d_{f(A)}\Delta_{m\times m}\circ d_Af.$$ Thus, the critical locus of $\Delta_{n\times n}$ contains the inverse image under $f$ of the critical locus of $\Delta_{m\times m}$. Since $m\geq 2$, in each case, the critical locus is the nonempty set of those matrices whose kernel has dimension $\geq 2$, this critical locus contains the origin, and this critical locus is irreducible of codimension $4$. Thus, the inverse image under $f$ of the critical locus of $\Delta_{m\times m}$ is nonempty (it contains the origin) and has codimension $\leq 4$ (since $\textbf{Mat}_{m\times m}$ is smooth). Since this is contained in the critical locus of $\Delta_{n\times n}$, and since the critical locus of $\Delta_{n\times n}$ is irreducible of codimension $4$, the inverse image of the critical locus of $\Delta_{m\times m}$ equals the critical locus of $\Delta_{n\times n}$. **QED**

Denote by $U_n\subset \text{Zero}(\Delta_{n\times n})$, resp. $U_m\subset \text{Zero}(\Delta_{m\times m})$, the open complement of the critical locus, i.e., the locus of matrices whose kernel has dimension precisely equal to $1$. By Lemma 1 and Lemma 2, $f$ restricts to an affine morphism $$f_U:U_n\to U_m.$$

**Proposition.** The cohomological dimension of sheaf cohomology for quasi-coherent sheaves on the quasi-affine scheme $U_n$ equals $2(n-1)$.

**Proof.**
The quasi-affine scheme $U_n$ admits a morphism, $$\pi_n:U_n \to \mathbb{P}^{n-1}\times (\mathbb{P}^{n-1})^*,$$ sending every singular $n\times n$ matrix $A$ parameterized by $U_n$ to the ordered pair of the kernel of $A$ and the image of $A$. The morphism $\pi_n$ is Zariski locally projection from a product, where the fiber is the affine group scheme $\textbf{GL}_{n-1}$. In particular, since $\pi_n$ is affine, the cohomological dimension of $U_n$ is no greater than the cohomological dimension of $\mathbb{P}^{n-1}\times (\mathbb{P}^{n-1})^*$. This equals the dimension $2(n-1)$.

More precisely, $U_n$ is simultaneously a principal bundle for both group schemes over $\mathbb{P}^{n-1}\times(\mathbb{P}^{n-1})^*$ that are the pullbacks via the two projections of $\textbf{GL}$ of the tangent bundle. Concretely, for a fixed $1$-dimension subspace $K$ of the $n$-dimensional vector space $V$ -- the kernel -- and for a fixed codimension $1$ subspace $I$ -- the image -- the set of invertible linear maps from $V/K$ to $I$ is simultaneously a principal bundle under precomposition by $\textbf{GL}(V/K)$ and a principal bundle under postcomposition by $\textbf{GL}(I)$. In particular, the pushforward of the structure sheaf, $$\mathcal{E}_n:=(\pi_n)_*\mathcal{O}_{U_n},$$ is a quasi-coherent sheaf that has an induced action of each of these group schemes.

The invariants for each of these actions is just $$\pi_n^\#:\mathcal{O}_{\mathbb{P}^{n-1}\times (\mathbb{P}^{n-1})^*}\to \mathcal{E}_n.$$ Concretely, the only functions on an algebraic group that are invariant under pullback by every element of the group are the constant functions. The group schemes and the principal bundle are each **Zariski** locally trivial. Consider the restriction of $\mathcal{E}_n$ on each open affine subset $U$ where the first group scheme is trivialized and the principal bundle is trivialized. The sections on this open affine give a $\mathcal{O}(U)$-linear representation of $\textbf{GL}_{n-1}$. Because $\textbf{GL}_{n-1}$ is **linearly** reductive, there is a *unique* splitting of this representation into its invariants, i.e., $\pi_n^\#\mathcal{O}(U)$, and a complementary representation (having trivial invariants and coinvariants). The uniqueness guarantees that these splittings glue together as we vary the trivializing opens. Thus, there is a splitting of $\pi_n^\#$ as a homomorphism of quasi-coherent sheaves, $$t_n:(\pi_n)_*\mathcal{O}_{U_n} \to \mathcal{O}_{\mathbb{P}^{n-1}\times (\mathbb{P}^{n-1})^*}.$$

For every invertible sheaf $\mathcal{L}$ on $\mathbb{P}^{n-1}\times (\mathbb{P}^{n-1})^*$, this splitting of $\mathcal{O}$-modules gives rise to a splitting, $$t_{n,\mathcal{L}}:(\pi_n)_*(\pi_n^*\mathcal{L})\to \mathcal{L}.$$ In particular, for every integer $q$, this gives rise to a surjective group homomorphism, $$H^q(t_{n,\mathcal{L}}):H^q(U_n,\pi_n^*\mathcal{L}) \to H^q(\mathbb{P}^{n-1}\times (\mathbb{P}^{n-1})^*,\mathcal{L}) .$$

Now let $\mathcal{L}$ be a dualizing invertible sheaf on $\mathbb{P}^{n-1}\times (\mathbb{P}^{n-1})^*.$ This has nonzero cohomology in degree $2(n-1)$. Thus, the cohomological dimension of sheaf cohomology for quasi-coherent $\mathcal{O}_{U_n}$-modules also equals $2(n-1)$. **QED**

Since $f_U$ is affine, the cohomological dimension for $U_n$ is no greater than the cohomological dimension for $U_m$. However, by the proposition, the cohomological dimension for $U_m$ equals $2(m-1)$. Since $1<m<n$, this is a contradiction.