On Fermat-type equations of signature $(r,r,p)$

Let $r\geq11$ be a prime. We show that there are infinitely many integers $C$ for which the Fermat-type equation $$ x^r+y^r=Cz^p $$ has no non-trivial primitive solutions for all sufficiently large (in terms of $r$ and~$C$) prime exponents~$p$. The proof uses several Frey curves to force simultaneou