In my understanding, weight is used to reweigh the losses from different classes (to avoid class-imbalance scenarios), rather than influencing the softmax logits.
Consider that the loss function is independent of softmax.
That is, In the cross-entropy loss function,
L_i(y, t) = -t_ij log y_ij (here t_ij=1)
y_i is the probability vector that can be obtained by any other way than just softmax. With weight, we are trying to reweigh L_i(y, t) based on the ground truth class.