Distribution of elements of cosets of small subgroups and applications

We obtain a series of estimates on the number of small integers and small order Farey fractions that belong to a given coset of a subgroup of order t of the group of units of the residue ring modulo a prime p, in the case when t is small compared to p. We give two applications of these results: to the simultaneous distribution of two high-degree monomials x ^k1 and x ^k2 modulo p and to a question of Holden and Moree on fixed points of the discrete logarithm.