Stable rational cohomology splitting of moduli of branched covers of curves

We consider the moduli stack ${\mathcal{H}}_{g,d}$ parametrising maps $\theta \text{\hspace{0.17em}⁣}:C\to L$ of degree $d$ between complex curves of genera $g$ and $0$. We provide a splitting of the stable rational cohomology of ${\mathcal{H}}_{g,d}$, for $g\to \mathrm{\infty }$, in terms of degrees of intermediate covers, and identify each direct summand in terms of factorisation homology. For $2\le d\le 5$, we further interpret the stable rational cohomology in terms of configuration spaces with labels in partial abelian monoids; we show in particular the existence of stable odd-dimensional cohomology classes for $d=4,5$.