Homomorphisms of graded modules
I will just take a guess that you mean $M$ is finitely generated, not finite.
Pick a set of homogeneous generators $g_1, \ldots, g_n$ of $M$ such that $g_i \in M_{m_i}$. For any $g_i$, $\varphi(g_i) \in \bigoplus_{j=1}^{k_m} N_{n_{i,j}}$, i.e., degrees seen by applying $\varphi$ to $g_i$ are $n_{i,j} - m_i$. Since $\varphi$ is completely determined by $g_i$, we know that $\varphi \in \sum_{i,j} \hbox{Hom}_{n_{i,j} - m_i}(M, N)$. (I use "sum" because there might be duplicates.)