In this article, we study bounds on the uniform distance between the cumulative distribution function of a standardized sum of independent centered random variables with moments of order four and its first-order Edgeworth expansion. Existing bounds are sharpened in two frameworks: when the variables are independent but not identically distributed and in the case of independent and identically distributed random variables. Improvements of these bounds are derived if the third moment of the distribution is zero. We also provide adapted versions of these bounds under additional regularity constraints on the tail behavior of the characteristic function. We finally present an application of our results to the lack of validity of one-sided tests based on the normal approximation of the mean for a fixed sample size.@en