@GeoffRobinson Thank you very much for your answer. Your references led me to a paper of Arad and Chillag (On Finite Groups with Conditions on the Centralizers of p-Elements, 1978) which discusses the exact same thing that I was looking for. The groups are called Cpp groups. For p=2, they are just CIT-groups and for p=3 are called Cθθ -groups.

Thank you very much. I think a similar argument shows that the converse is also true, i.e. $H \cap K = 1$ for some $g \in G$ iff $r \ne 2s$.

@DerekHolt For example if $r=4$ and $s=2$ then for all $g \in G$ the subgroup $M^g \cap M$ is non-trivial. It seems true whenever $r=2s$ but I have not seen any proof of that.

@Carnahan The latter assertion looks incorrect. Note that this forces $C_x (G)$ to be normal in $G$.

@ Tobias If you take G = A4, the alternating group of order 12, then it satisfies the above hypotheses . However, A4 is not supersolvable. Your assertion is true with the additional condition that Z(G) > 1.

@Jack Thank you very much for references.