We introduce the $C_{p^n}$-Mahowald invariant: a relation $\pi_\star \smash{S_{\raisemath{2pt}{C_{p^{n-1}}}}}\rightharpoonup \pi_\ast S$ between the equivariant and classical stable stems which reduces to the classical Mahowald invariant when $n=1$. We compute the $C_{p^n}$-Mahowald invariants of all elements in the Burnside ring $\smash{A(C_{p^{n-1}}) = \pi_0S_{\raisemath{2pt}{C_{p^{n-1}}}}}$, extending Mahowald and Ravenel's computation of $M_{C_p}(p^k)$. As a consequence, we determine the image of the $C_p$-geometric fixed point map $\Phi^{C_p} \colon \pi_V S_{\raisemath{2pt}{C_{p^{n}}}}\rightarrow \pi_0 \smash{S_{\raisemath{2pt}{C_{p^{n}}/C_p}}}\cong A(C_{p^{n-1}})$ when $V$ is fixed point free, extending classical theorems of Bredon, Landweber, and Iriye for $n=1$.
Our work exhibits patterns suggesting an equivariant Adams periodicity. We refine this to produce equivariant lifts of Adams' $v_1$-self maps on Moore spectra.