We study the Mackey-Glass type monostable delayed reaction diffusion equation with a unimodal birth function g(u). This model, designed to describe evolution of single species populations, is considered here in the presence of the weak Allee effect (g(u0) > g'(0)u0 for some u0 > 0). We focus our attention on the existence of slow monotonic traveling fronts to the equation: under given assumptions, this problem seems to be rather difficult since the usual positivity and monotonicity arguments are not effective. First, we solve the front existence problem for small delays, h is an element of [0, hp], where hp, given by an explicit formula, is optimal in a certain sense. Then we take a representative piece-wise linear unimodal birth function which makes possible explicit computation of traveling fronts. In this case, we find out that a) increase of delay can destroy asymptotically stable pushed fronts; b) the set of all admissible wavefront speeds has usual structure of a semi-infinite interval [c*, +infinity); c) for each h >= 0, the pushed wavefront is unique (if it exists); d) pushed wave can oscillate slowly around the positive equilibrium for sufficiently large delays.