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.

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.

Not sure if you're already aware of the exact sequence which implies Hurewicz, but we have $$H_{i+1}(X)\to H_{i+1}(\pi)\to (H_i(X))_\pi\to H_i(X)\to H_i(\pi)\to0$$ where $H_*(\pi)$ is discrete group cohomology and $\pi$ acts on $H_i(X)$ via the universal cover over $X$ (I mean that's one way to phrase the action). See classic book of Spanier, or can be proved by spectral sequence, c.f. Ken Brown's bible "Cohomology of Groups" (exercise IV.5.1 and exercise VII.7.6).

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.