Breaking the 6/5 threshold for sums and products modulo a prime

Let $A\subset {\mathbb{F}}_p$ of size at most $p^{3\mathrm{/}5}$. We show $$\mathrm{\mid }A+A\mathrm{\mid }+\mathrm{\mid }AA\mathrm{\mid }\gtrsim \mathrm{\mid }A{\mathrm{\mid }}^{6\mathrm{/}5+c},$$ for $c=4\mathrm{/}305$. Our main tools are the cartesian product point–line incidence theorem of Stevens and de Zeeuw and the theory of higher energies developed by the second author.