  Mantel's theorem states that every $n$-vertex graph with $\lfloor
\frac{n^2}{4} \rfloor +t$ edges, where $t>0$, contains a triangle. The problem
of determining the minimum number of triangles in such a graph is usually
referred to as the Erd\H{o}s-Rademacher problem. Lov\'asz and Simonovits proved
that there are at least $t\lfloor n/2 \rfloor$ triangles in each of those
graphs. Katona and Xiao considered the same problem under the additional
condition that there are no $s-1$ vertices covering all triangles. They settled
the case $t=1$ and $s=2$. Solving their conjecture, we determine the minimum
number of triangles for every fixed pair of $s$ and $t$, when $n$ is
sufficiently large. Additionally, solving another conjecture of Katona and
Xiao, we extend the theory for considering cliques instead of triangles.
