On the fourth power mean of the analogous general Kloosterman sum (Q897546)

The authors study generalized Kloosterman sums \[ C(m,n,k,\chi;q)= \sum_{a=1}^{q}\chi(a)e\left(\frac{m\bar a^k+na}{q }\right), \] where \(q\), \(m\), \(n\), \(k\) are given positive integers, \(q \geq 3\), \(e(x)=e^{2\pi i x}\), \(a\bar a\equiv 1\pmod{q}\) and \(\chi\) is a character \(\mod{q}\). The main result of the paper is an explicit formula for the sum \[ \sum_{\chi\mod q}\sum_{{m=1\atop(m,q)=1}}^{q}|C(m,n,k,\chi;q)|^4. \]