  We prove non-trivial bounds for general bilinear forms in hyper-Kloosterman
sums when the sizes of both variables may be below the P\'olya-Vinogradov
range. We then derive applications to the second moment of holomorphic cusp
forms twisted by characters modulo primes, and to the distribution in
arithmetic progressions to large moduli of certain Eisenstein-Hecke
coefficients on $\GL_3$. Our main tools are new bounds for certain complete
sums in three variables over finite fields, proved using methods from algebraic
geometry, especially $\ell$-adic cohomology and the Riemann Hypothesis.
