  The electromagnetic counterparts to gravitational wave (GW) merger events
hold immense scientific value, but are difficult to detect due to the typically
large localisation errors associated with GW events. The Low-Frequency Array
(LOFAR) is an attractive GW follow-up instrument owing to its high sensitivity,
large instantaneous field of view, and ability to automatically trigger on
events to probe potential prompt emission within minutes. Here, we report on
144-MHz LOFAR radio observations of three GW merger events containing at least
one neutron star that were detected during the third GW observing run.
Specifically, we probe 9 and 16 per cent of the location probability density
maps of S190426c and S200213t, respectively, and place limits at the location
of an interesting optical transient (PS19hgw/AT2019wxt) found within the
localisation map of S191213g. While these GW events are not particularly
significant, we use multi-epoch LOFAR data to devise a sensitive wide-field GW
follow-up strategy to be used in future GW observing runs. In particular, we
improve on our previously published strategy by implementing direction
dependent calibration and mosaicing, resulting in nearly an order of magnitude
increase in sensitivity and more uniform coverage. We achieve a uniform
$5\sigma$ sensitivity of $870$ $\mu$Jy across a single instantaneous LOFAR
pointing's 21 deg$^{2}$ core, and a median sensitivity of 1.1 mJy when
including the full 89 deg$^{2}$ hexagonal beam pattern. We also place the
deepest transient surface density limits yet on of order month timescales for
surveys between 60--340 MHz (0.017 deg$^{-2}$ above $2.0$ mJy and 0.073
deg$^{-2}$ above $1.5$ mJy).
