  We investigate a model of continuous-time simple random walk paths in
$\mathbb{Z}^d$ undergoing two competing interactions: an attractive one towards
the large values of a random potential, and a self-repellent one in the spirit
of the well-known weakly self-avoiding random walk. We take the potential to be
i.i.d.~Pareto-distributed with parameter $\alpha>d$, and we tune the strength
of the interactions in such a way that they both contribute on the same scale
as $t\to\infty$.
  Our main results are (1) the identification of the logarithmic asymptotics of
the partition function of the model in terms of a random variational formula,
and, (2) the identification of the path behaviour that gives the overwhelming
contribution to the partition function for $\alpha>2d$: the random-walk path
follows an optimal trajectory that visits each of a finite number of random
lattice sites for a positive random fraction of time. We prove a law of large
numbers for this behaviour, i.e., that all other path behaviours give strictly
less contribution to the partition function. The joint distribution of the
variational problem and of the optimal path can be expressed in terms of a
limiting Poisson point process arising by a rescaling of the random potential.
The latter convergence is in distribution and is in the spirit of a standard
extreme-value setting for a rescaling of an i.i.d. potential in large boxes,
like in \cite{KLMS09}.
