Summary: We prove new \(\ell^p(\mathbb{Z}^d)\) bounds for discrete spherical averages in dimensions \(d\geq 5\). We focus on the case of lacunary radii, first for general lacunary radii, and then for certain kinds of highly composite choices of radii. In particular, if \(A_\lambda f\) is the spherical average of \(f\) over the discrete sphere of radius \(\lambda\), we have \[ \|\mathrm{sup}_k|A_{\lambda_{k}}|\|_{\ell^{p}(\mathbb{Z}^{d})}, \quad \frac{d-2}{d-3}<p\leq\frac{d}{d-2},\, d\geq 5, \] for any lacunary sets of integers \(\{\lambda^2_k\}\). We follow a style of argument from our prior paper [“An endpoint sparse bound for the discrete spherical maximal functions”, Preprint, arXiv:1810.02240], addressing the full supremum. The relevant maximal operator is decomposed into several parts; each part requires only one endpoint estimate.