  We obtain a new estimate for Kloosterman sum with primes $p\leqslant X$ to
composite modulo $q$, that is, for the exponential sum of the type \[
\sum\limits_{p\leqslant X,\;p\,\nmid q}\exp{\biggl(\frac{2\pi
i}{q}\bigl(a\overline{p}+bp\bigr)\,\biggr)},\quad (ab,q)=1,\quad
p\overline{p}\equiv 1\pmod{q}, \] which is non-trivial in the case when
$q^{\,3/4+\varepsilon}\leqslant X\ll q^{\,3/2}$. We also apply this estimate to
the proof of solvability of some congruences with inverse prime residues
$\pmod{q}$.
