  The problem of random walk is considered in one dimension in the simultaneous
presence of a quenched random force field and long-range connections the
probability of which decays with the distance algebraically as p_l ~ \beta
l^{-s}. The dynamics are studied mainly by a numerical strong disorder
renormalization group method. According to the results, for s>2 the long-range
connections are irrelevant and the mean-square displacement increases as
<x^2(t)> ~ (ln t)^{2/\psi} with the barrier exponent \psi=1/2, which is known
in one-dimensional random environments. For s<2, instead, the quenched disorder
is found to be irrelevant and the dynamical exponent is z=1 like in a
homogeneous environment. At the critical point, s=2, the interplay between
quenched disorder and long-range connections results in activated scaling,
however, with a non-trivial barrier exponent \psi(\beta), which decays
continuously with \beta, but is independent of the form of the quenched
disorder. Upper and lower bounds on \psi(\beta) are established and numerical
estimates are given for various values of \beta. Beside random walks, accurate
numerical estimates of the graph dimension and the resistance exponent are
given for various values of \beta at s=2.
