The reduction you require is a (very) special case of Marsden-Weinstein (1974). Your one constant of the motion, say $\psi$, is the moment map of an action of the additive group $G=\mathbf R$ on the symplectic manifold with coordinates $(p_1,q_1,\dots,p_n,q_n)$ — namely $\psi$’s hamiltonian flow, obtained by solving $$ \frac{d}{dt}\begin{pmatrix}p_i\\q_i\end{pmatrix}= \begin{pmatrix}-\partial\psi/\partial q_i\\\phantom{-}\partial\psi/\partial p_i\end{pmatrix}. \tag1 $$ Their Theorem 1 says that each level $\psi^{-1}(\mu)$ is a coisotropic submanifold whose null leaves are the $G$-orbits, so the symplectic form descends to the leaf space $\psi^{-1}(\mu)\,/\,G$. This is the reduced space, of dimension 2(n-1). Their Theorem 2 adds that $H$, being constant on leaves since $\{H,\psi\}=0$, descends to a function $H_\mu$ on the reduced space. This is the reduced system.
(This is all subject to technical conditions: (1) complete, $\mu$ “weakly regular” value of $\psi$, $G$-action on levels free and proper — which I don’t think matter much in your local formulation. E.g. taking $\psi$ as $q_n$, the quotient means “ignore $p_n$”, and Darboux charts on the symplectic manifold $\smash{\psi^{-1}(\mu)\,/\,G}$ give your desired $(p_1,q_1,\dots,p_{n-1},q_{n-1})$.)
As their introduction points out, this special case $G=\mathbf R$ had long been known as the theory of “ignorable coordinates”, exposed with plenty of examples in e.g. Whittaker (1904, §38 sq). Other nice example from Souriau, who had the theory for abelian $G$ (1970, Chap. III, 12.153 sq): at a negative level of a hydrogen atom’s energy $\psi$, the reduced space is $\smash{\mathrm S^2\times\mathrm S^2}$ and components of angular momentum and the eccentricity (a.k.a. Lenz) vector descend to generate an $\mathrm{SO}(4)$ action there.