Maximizing the number of cliques in $K_{r+1}$-free graphs with forbidden properties

Ferrero and Lesniak in 2018 found the maximum numbers of edges in $r$-partite non-Hamiltonian graphs. Recently we found the maximum numbers of edges and $t$-cliques in $K_{r+1}$-free graphs (1) that are not Hamiltonian or (2) that satisfy a condition on low-degree vertices related to Pósa's theorem.