The low-temperature thermoelectric resonant transport through the Anderson impurities under the Kondo resonance conditions is studied. It is shown that for the smooth Anderson impurity (AI) density of states the thermoelectric current is parametrically larger than that in the absence of on-site Coulomb correlation, which is the signature of the Kondo effect. In the case of singular AI density of states some new characteristic temperature appears. For temperatures smaller than this, the Kondo-type contributions to the thermoelectric current are important, while in the opposite case the conventional picture of thermoelectric transport returns.