Kloosterman sums with primes to composite moduli

We obtain a new estimate for Kloosterman sum with primes $p⩽X$ to composite modulo $q$, that is, for the exponential sum of the type $$\sum _{p⩽X,p\nmid q}\exp \left(\frac{2\pi i}{q}\right(a\bar{p}+bp\left)\right),(ab,q)=1,p\bar{p}\equiv 1(\mathrm{m}\mathrm{o}\mathrm{d}q),$$ which is non-trivial in the case when $q^{3\mathrm{/}4+\epsilon }⩽X\ll q^{3\mathrm{/}2}$. We also apply this estimate to the proof of solvability of some congruences with inverse prime residues $(\mathrm{m}\mathrm{o}\mathrm{d}q)$.