Given a symmetric $n\times n$ matrix $P$ with $0 \le P(u, v)\le 1$, we define a random graph $G_{n, P}$ on $[n]$ by independently including any edge $\{u, v\}$ with probability $P(u, v)$. For $k\ge 1$ let $\mathcal{A}_k$ be the property of containing $\lfloor k/2 \rfloor$ Hamilton cycles, and one perfect matching if $k$ is odd, all edge-disjoint. With an eigenvalue condition on $P$, and conditions on its row sums, $G_{n, P}\in \mathcal{A}_k$ happens with high probability if and only if $G_{n, P}$ has minimum degree $k$ whp. We also provide a hitting time version. As a special case, the random graph process on pseudorandom $(n, d, μ)$-graphs with $μ\le d(d/n)^α$ for some constant $α> 0$ has property $\mathcal{A}_k$ as soon as it acquires minimum degree $k$ with high probability.