  We prove new incidence bounds between a plane point set, which is a Cartesian
product, and a set of translates $H$ of the hyperbola $xy=\lambda\neq 0$, over
a field of asymptotically large positive characteristic $p$. They improve
recent bounds by Shkredov, which are based on using explicit incidence
estimates in the early terminated procedure of repeated applications of the
Cauchy-Schwarz inequality, underlying many qualitative results related to
growth and expansion in groups. The improvement -- both quantitative, plus we
are able to deal with a general $H$, rather than a Cartesian product -- is
mostly due to a non-trivial "intermediate" bound on the number of $k$-rich
M\"obius hyperbolae in positive characteristic. In addition, we make an
observation that a certain energy-type quantity in the context of $H$ can be
bounded via the $L^2$-moment of the Minkowski distance in $H$ and can therefore
fetch the corresponding estimates apropos of the Erd\H{o}s distinct distance
problem.
