Incomplete Kloosterman sums and solvability of some congruences with prime variables

The talk is devoted to a new estimate for a special incomplete Kloosterman sum and its applications to the problems of the solvability of some congruences modulo $q$. The variables in these congruences run through prime numbers lying in the short interval of the type $(1,N]$, where $N\ll q^{1-c}$, $c>0$.