Mapping spaces and homology isomorphisms

Let $\operatorname{Map}(K,X)$ denote the space of pointed continuous maps from a finite cell complex $K$ to a space $X$. Let $E_*$ be a generalized homology theory. We use Goodwillie calculus methods to prove that under suitable conditions on $K$and $X$, $\operatorname{Map}(K, X)$ will send an $E_*$-isomorphism in either variable to a map that is monic in $E_*$ homology. Interesting examples arise by letting $E_*$ be $K$-theory, the finite complex $K$ be a sphere, and the map in the $X$ variable be an exotic unstable Adams map between Moore spaces.