Motivic cycles on K3 double covers of del Pezzo surfaces

We construct motivic cohomology cycles in the group $H_{\mathcal{M}}^3(Z,\mathbb{Q}(2))$ where $Z$ is a K3 surface obtained as a double cover of a del Pezzo surface $X$ branched at a curve in $\mathrm{\mid }-2K_X\mathrm{\mid }$. The construction uses (-1) curves on the del Pezzo and is a generalization of a recent pre-print of Ken Sato arXiv: 2408.09102 where he considers the case of fourfold covers of ${\mathbb{P}}^2$ branched at a quartic curve.