  Results are presented of searches for continuous gravitational waves from 20
accreting millisecond X-ray pulsars with accurately measured spin frequencies
and orbital parameters, using data from the third observing run of the Advanced
LIGO and Advanced Virgo detectors. The search algorithm uses a hidden Markov
model, where the transition probabilities allow the frequency to wander
according to an unbiased random walk, while the $\mathcal{J}$-statistic
maximum-likelihood matched filter tracks the binary orbital phase. Three narrow
sub-bands are searched for each target, centered on harmonics of the measured
spin frequency. The search yields 16 candidates, consistent with a false alarm
probability of 30% per sub-band and target searched. These candidates, along
with one candidate from an additional target-of-opportunity search done for SAX
J1808.4$-$3658, which was in outburst during one month of the observing run,
cannot be confidently associated with a known noise source. Additional
follow-up does not provide convincing evidence that any are a true
astrophysical signal. When all candidates are assumed non-astrophysical, upper
limits are set on the maximum wave strain detectable at 95% confidence,
$h_0^{95\%}$. The strictest constraint is $h_0^{95\%} = 4.7\times 10^{-26}$
from IGR J17062$-$6143. Constraints on the detectable wave strain from each
target lead to constraints on neutron star ellipticity and $r$-mode amplitude,
the strictest of which are $\epsilon^{95\%} = 3.1\times 10^{-7}$ and
$\alpha^{95\%} = 1.8\times 10^{-5}$ respectively. This analysis is the most
comprehensive and sensitive search of continuous gravitational waves from
accreting millisecond X-ray pulsars to date.
