= Solution
Take a based <CW complex> model for $X$. Since it is $(n-1)$-connected and $n\ge2$, its reduced ordinary <homology> vanishes below degree $n$, with arbitrary constant coefficients. This also follows from a <CW complex> model with no non-basepoint cells below degree $n$. The <connective spectrum> $E$ has $\pi_qE=0$ for $q<0$.
Filter $X$ by its skeleta. The quotients of consecutive stages are wedges of spheres, so the exact sequences in <represented homology theory> give the <homological Atiyah-Hirzebruch spectral sequence>
$$
E^2_{p,q}=\widetilde H_p(X;\pi_qE)\Longrightarrow\widetilde E_{p+q}(X),\qquad d_r:E^r_{p,q}\longrightarrow E^r_{p-r,q+r-1}.
$$
It is first quadrant because $E$ is a <connective spectrum>. In a fixed total degree it converges with a finite filtration: cells of sufficiently high dimension cannot contribute in that degree or affect it through a boundary, by the same connectivity bound.
In total degree $n$, every term with $p<n$ is zero, and terms with $q<0$ are zero. The only possible term is
$$
E^2_{n,0}=\widetilde H_n(X;\pi_0E)=H_n(X;\mathbb Z).
$$
No differential can leave it: the target has spatial degree $n-r<n$. No differential can enter it: its source would be $(n+r,1-r)$, whose coefficient degree is negative for every $r\ge2$. Thus it survives unchanged and is the only filtration quotient of $\widetilde E_n(X)$; there is no extension problem. The ordinary <Hurewicz theorem> now yields
$$
\boxed{\widetilde E_n(X)\cong H_n(X;\mathbb Z)\cong\pi_n(X).}
$$
The isomorphism uses the stated identification $\pi_0E\cong\mathbb Z$ and is natural in $X$. This is the <bottom-degree generalized Hurewicz isomorphism>.
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