Suppose that for $A\subset \omega$ there is a partial recursive function $\psi$ such that when the domain of the $e$th p.r. function ($W_e$) does not meet $A$, $\psi(e)\downarrow \notin A\cup W_e$. Apparently this implies that $A$ is not simple, but I am having trouble proving it.
My first thought is that if $A$ is coinfinite and r.e. then you could construct an infinite r.e. set not meeting $A$ by taking the image of the set $\{e: W_e = [0,n]\setminus A\}$ (or some subset of this)...but I don't think that this is r.e. and I don't see a way to extract an infinite r.e. subset from it.