Not sure if you're already aware of an exact sequence which implies Hurewicz's theorem, but if $X$ is a space with $\pi_i(X)=0$ for $1<i<n$ then we have
$$H_{n+1}(X)\to H_{n+1}(\pi)\to \pi_n(X)_{\pi}\to H_n(X)\to H_n(\pi)\to0$$ where $\pi=\pi_1(X)$ is the fundamental group and $H_*(\pi)$ is discrete group cohomology, and $\pi_n(X)_{\pi}$ denotes the coinvariants. See e.g. Ken Brown "Cohomology of Groups"  exercise VII.7.6.