The conductance G and the thermopower S of a saddle-point-potential quantum point contact in the presence of a δ-potential impurity are calculated. For the case when an attractive impurity is located inside the quantum point contact, there are dips, peaks, and kinks in G. These structures can be found below and near the band bottom of some transverse subbands, respectively. The peaks in G give rise to double peaks in S and the dips in G give rise to a shift in the peak positions of S, towards a smaller chemical potential value. In addition, a broad dip in G is found to give rise to negative thermopower in regions between the peaks of S. For the case when an attractive impurity is located in the classical forbidden region for the electrons, we find in G the structures that correspond to resonance tunneling and resonance reflection. The corresponding S is found to show large and negative spikes. Our study shows that structures not so transparent in G manifest unequivocally in S, rendering S a very informative physical quantity to be measured.