  A {\em generalized hyperfocused arc} $\mathcal H $ in $PG(2,q)$ is an arc of
size $k$ with the property that the $k(k-1)/2$ secants can be blocked by a set
of $k-1$ points not belonging to the arc. We show that if $q$ is a prime and
$\mathcal H$ is a generalized hyperfocused arc of size $k$, then $k=1,2$ or 4.
Interestingly, this problem is also related to the (strong) cylinder conjecture
[Ball S.: The polynomial method in Galois geometries, in Current research
topics in Galois geometry, Chapter 5, Nova Sci. Publ., New York, (2012)
105-130], as we point out in the last section.
