  We obtain a new bound for the number of solutions to polynomial equations in
cosets of multiplicative subgroups in finite fields, which generalises previous
results of P. Corvaja and U. Zannier (2013). We also obtain a conditional
improvement of recent results of J. Bourgain, A. Gamburd and P. Sarnak (2016)
and S. V. Konyagin, S. V. Makarychev, I. E. Shparlinski and I. V. Vyugin (2019)
on the structure of solutions to the reduction of the Markoff equation $x^2 +
y^2 + z^2 = 3 x yz$ modulo a prime $p$.
