  The solutions of the perturbed first Painlev\'e equation $y"=6y^2-x^\mu$,
$\mu>-4$, are uniquely determined by the free constant $C$ multiplying the
exponentially small terms in the complete large $x$ asymptotic expansions. Full
details are given, including the nonlinear Stokes phenomenon, and the
computation of the relevant Stokes multipliers. We derive asymptotic
approximations, depending on $C$, for the locations of the singularities that
appear on the boundary of the sectors of validity of these
exponentially-improved asymptotic expansions. Several numerical examples
illustrate the power of the approximations. For the tri-tronqu\'ee solution of
the unperturbed first Painlev\'e equation we give highly accurate numerics for
the values at the origin and the locations of the zeros and poles.
