  Recent experiments have measured the signatures of the Kondo effect in the
zero-field thermopower of strongly correlated quantum dots [Svilans {\em et
al.,} Phys. Rev. Lett. {\bf 121}, 206801 (2018); Dutta {\em et al.,} Nano Lett.
{\bf 19}, 506 (2019)]. They confirm the predicted Kondo-induced sign change in
the thermopower, upon increasing the temperature through a gate-voltage
dependent value $T_{1}\gtrsim T_{\rm K}$, where $T_{\rm K}$ is the Kondo
temperature. Here, we use the numerical renormalization group (NRG) method to
investigate the effect of a finite magnetic field $B$ on the thermopower of
such quantum dots. We show that, for fields $B$ exceeding a gate-voltage
dependent value $B_{0}$, an additional sign change takes place in the Kondo
regime at a temperature $T_{0}(B\geq B_{0})>0$ with $T_0<T_1$. The field
$B_{0}$ is comparable to, but larger than, the field $B_{c}$ at which the
zero-temperature spectral function splits in a magnetic field. The validity of
the NRG results for $B_{0}$ are checked by comparison with asymptotically exact
higher-order Fermi-liquid calculations [Oguri {\em et al.,} Phys. Rev. B {\bf
97}, 035435 (2018)]. Our calculations clarify the field-dependent signatures of
the Kondo effect in the thermopower of Kondo-correlated quantum dots and
explain the recently measured trends in the $B$-field dependence of the
thermoelectric response of such systems [Svilans {\em et al.,} Phys. Rev. Lett.
{\bf 121}, 206801 (2018)].
