Take a symmetric Nakayama algebra with Kupisch series [3,3]. (see for example the book by assem simson skowronski in the chapter about nakayama algebras for the meaning) Then $Hom(e_0 A, soc(e_1A))=Hom(e_0 A, S_1)=0$, but $Hom(e_0A , e_1A)$ is nonzero and thus has nonzero socle.

Take a symmetric Nakayama algebra with Kupisch series [3,3]. (see for example the book by assem simson skowronski in the chapter about nakayama algebras for the meaning) Then $Hom(e_0 A, soc(e_1A))=Hom(e_0 A, S_1)=0$, but $Hom(e_0A , e_1A)$ is nonzero and thus has nonzero socle.

Next try: Let $A=K[x]/(x^2)$ with simple modules $S$ and $N=A \oplus S$ and $M=S$. Then $Hom(N,rad(M))=0$. But $Hom(N,M)$ is a projective indecomposable $End(N)$-module which is not simple and thus has nonzero radical.