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
$$H_{n+1}(X)\to H_{n+1}(\pi)\to \pi_n(X)_{\pi}\to H_n(X)\stackrel{\psi}{\to} H_n(\pi)\to0$$ 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.

Generally, without assumptions on the homotopy groups, that map $\psi$ exists (in all degrees) and is canonical: You take a projective resolution of $\mathbb Z$ over $\mathbb Z\pi$, and you take the complex of free $\mathbb Z\pi$-modules $C_*(\text{universal cover of }X)$, to build a canonical chain map between those resolutions that induces $\psi$. This is also explained in the above reference somewhere in Chapter II.