  During the first half of the fourth observing run (O4a) of the International
Gravitational Wave Network (IGWN), the Zwicky Transient Facility (ZTF)
conducted a systematic search for kilonova (KN) counterparts to binary neutron
star (BNS) and neutron star-black hole (NSBH) merger candidates. Here, we
present a comprehensive study of the five high-significance (FAR < 1 per year)
BNS and NSBH candidates in O4a. Our follow-up campaigns relied on both
target-of-opportunity observations (ToO) and re-weighting of the nominal survey
schedule to maximize coverage. We describe the toolkit we have been developing,
Fritz, an instance of SkyPortal, instrumental in coordinating and managing our
telescope scheduling, candidate vetting, and follow-up observations through a
user-friendly interface. ZTF covered a total of 2841 deg$^2$ within the skymaps
of the high-significance GW events, reaching a median depth of g~20.2 mag. We
circulated 15 candidates, but found no viable KN counterpart to any of the GW
events. Based on the ZTF non-detections of the high-significance events in O4a,
we used a Bayesian approach, nimbus, to quantify the posterior probability of
KN model parameters that are consistent with our non-detections. Our analysis
favors KNe with initial absolute magnitude fainter than -16 mag. The joint
posterior probability of a GW170817-like KN associated with all our O4a
follow-ups was 64%. Additionally, we use a survey simulation software,
simsurvey, to determine that our combined filtered efficiency to detect a
GW170817-like KN is 36%, when considering the 5 confirmed astrophysical events
in O3 (1 BNS and 4 NSBH), along with our O4a follow-ups. Following Kasliwal et
al. (2020), we derived joint constraints on the underlying KN luminosity
function based on our O3 and O4a follow-ups, determining that no more than 76%
of KNe fading at 1 mag/day can peak at a magnitude brighter than -17.5 mag.
