The aim of the paper is to present a test of goodness of fit with weigths in the classes based on weighted $\left( h,\phi \right) $-divergences. This family of divergences generalizes in some sense the previous weighted divergences studied by [O. Frank, M.L. Menéndez and L. Pardo: Asymptotic distributions of weighted divergence between discrete distributions. Comm. Statist. - Theory Methods 27 (1998), 4, 867-885.] and [J.N. Kapur: Measures of Information and their Applications. Wiley, New York 1994.]. The weighted $\left( h,\phi \right)$-divergence between an empirical distribution and a fixed distribution is here investigated for large simple random samples, and the asymptotic distributions are shown to be either normal or equal to the distribution of a linear combination of independent chi-square variables. Some approximations to the linear combination of independent chi-square variables are presented.