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In the paper, we establish a new estimate for Kloosterman sum over primeswith respect to an arbitrary modulus $q$. This estimate together with somerecent results of the second author are applied to the problem of solvabilityof the congruence \[ g(p_{1})\, + \,\ldots\, + \,g(p_{k}) \,\equiv\, m\pmod{q}\] in prime variables $p_{1},\ldots, p_{k}\le N$, $N\le q^{1-\delta}$. Here$g(x)\,\equiv\,a\overline{x}+bx\pmod{q}$, where $a,b,m$ and $k\ge 3$ arearbitrary integers, $(ab,q)=1$. The main result of the paper gives anasymptotic formula for the number of solutions for the case when $q$ is coprimeto $6$.  In this version, we correct some typos and small errors.

Kloosterman sums with primes and solvability of one congruence with inverse residues. I