  In 2011, Fang et al. in (J. Combin. Theory A 118 (2011) 1039-1051) posed the
following problem: Classify non-normal locally primitive Cayley graphs of
finite simple groups of valency $d$, where either $d\leq 20$ or $d$ is a prime
number. The only case for which the complete solution of this problem is known
is of $d=3$. Except this, a lot of efforts have been made to attack this
problem by considering the following problem: Characterize finite nonabelian
simple groups which admit non-normal locally primitive Cayley graphs of certain
valency $d\geq4$. Even for this problem, it was only solved for the cases when
either $d\leq 5$ or $d=7$ and the vertex stabilizer is solvable. In this paper,
we make crucial progress towards the above problems by completely solving the
second problem for the case when $d\geq 11$ is a prime and the vertex
stabilizer is solvable.
