  Let $\Gamma\subset \mathbb{R}^2$ be a simple closed curve which is smooth
except at the origin, at which it has a power cusp and coincides with the curve
$|x_2|=x_1^p$ for some $p>1$. We study the eigenvalues
  of the Schr\"odinger operator $H_\alpha$ with the attractive
$\delta$-potential of strength $\alpha>0$ supported by $\Gamma$, which is
defined by its quadratic form \[ H^1(\mathbb{R}^2)\ni u\mapsto
\iint_{\mathbb{R}^2} |\nabla u|^2\,\mathrm{d}x-\alpha\int_\Gamma u^2\,
\mathrm{d}s, \] where $\mathrm{d}s$ stands for the one-dimensional Hausdorff
measure on $\Gamma$. It is shown that if $n\in\mathbb{N}$ is fixed and $\alpha$
is large, then the well-defined $n$th eigenvalue $E_n(H_\alpha)$ of $H_\alpha$
behaves as \[ E_n(H_\alpha)=-\alpha^2 + 2^{\frac{2}{p+2}} \mathcal{E}_n
\,\alpha^{\frac{6}{p+2}} + \mathcal{O}(\alpha^{\frac{6}{p+2}-\eta}), \] where
the constants $\mathcal{E}_n>0$ are the eigenvalues of an explicitly given
one-dimensional Schr\"odinger operator determined by the cusp, and $\eta>0$.
Both main and secondary terms in this asymptotic expansion are different from
what was observed previously for the cases when~$\Gamma$ is smooth or piecewise
smooth with non-zero angles.
