Is there any way to simplify this matrix expression, using something like a distributive property?
$A \otimes \left( M_1 \oplus M_2 \right)$
where $\otimes$ is the Kronecker product and $\oplus$ is the direct sum. We're assuming all of the matrix dimensions work out.
I guess to some extent you can do this:
For matrices, one can write the direct sum as $M_1 \oplus M_2 = \begin{pmatrix} M_1 & 0 \\ 0 & M_2 \end{pmatrix} = M$. Now assuming the standard tensor product for matrices and labeling $ A = \begin{pmatrix} A_{11} & A_{12} \\ A_{21} & A_{21} \end{pmatrix}$ we find $ (M_1 \oplus M_2)\otimes A \ = \begin{pmatrix} M_1 & 0 \\ 0 & M_2 \end{pmatrix}\otimes \begin{pmatrix} A_{11} & A_{12} \\ A_{21} & A_{21} \end{pmatrix} = \begin{pmatrix} M_1 \otimes A & 0 \\ 0 & M_2 \otimes A\end{pmatrix} = (M_1\otimes A) \oplus (M_2 \otimes A)$.
Now note that I have switched the direction from which to tensor by $A$ the other direction does not look as lovely however can be found by explicitly writing down the product, and one finds that the elements of the resulting matrix are identical up to moving the entries around a bit. Now depending on what you actually want to use this for you could fall back on the universal property of the tensor product and choose the definition where your direction works out to have nicer notation.