Generalized Turán results for disjoint cliques

The generalized Turán number $\mathrm{e}\mathrm{x}(n,H,F)$ is the largest number of copies of $H$ in $n$-vertex $F$-free graphs. We denote by $tF$ the vertex-disjoint union of $t$ copies of $F$. Gerbner, Methuku and Vizer in 2019 determined the order of magnitude of $\mathrm{e}\mathrm{x}(n,K_s,t{K}_r)$. We extend this result in three directions. First, we determine $\mathrm{e}\mathrm{x}(n,K_s,t{K}_r)$ exactly for sufficiently large $n$. Second, we determine the asymptotics of the analogous number for $p$-uniform hypergraphs. Third, we determine the order of magnitude of $\mathrm{e}\mathrm{x}(n,H,t{K}_r)$ for every graph $H$, and also of the analogous number for $p$-uniform hypergraphs.