  We show that various loci of stable curves of sufficiently large genus
admitting degree $d$ covers of positive genus curves define non-tautological
algebraic cycles on $\overline{\mathcal{M}}_{g,N}$, assuming the non-vanishing
of the $d$-th Fourier coefficient of a certain modular form. Our results build
on those of Graber-Pandharipande and van Zelm for degree 2 covers of elliptic
curves; the main new ingredient is a method to intersect the cycles in question
with boundary strata, as developed recently by Schmitt-van Zelm and the author.
