  This paper investigates the frequency of transiting planetary systems around
metal-polluted white dwarfs using high-cadence photometry from ULTRACAM and
ULTRASPEC on the ground, and space-based observations with TESS. Within a
sample of 313 metal-polluted white dwarfs with available TESS light curves, two
systems known to have irregular transits are blindly recovered by
box-least-squares and Lomb-Scargle analyses, with no new detections, yielding a
transit fraction of 0.8 (-0.4, +0.6) per cent. Planet detection sensitivities
are determined using simulated transit injection and recovery for all light
curves, producing upper limit occurrences over radii from dwarf to Kronian
planets, with periods from 1 h to 27 d. The dearth of short-period, transiting
planets orbiting polluted white dwarfs is consistent with engulfment during the
giant phases of stellar evolution, and modestly constrains dynamical
re-injection of planets to the shortest orbital periods. Based on simple
predictions of transit probability, where (R + Rp)/a ~ 0.01, the findings here
are nominally consistent with a model where 100 per cent of polluted white
dwarfs have circumstellar debris near the Roche limit; however, the small
sample size precludes statistical confidence in this result. Single transits
are also ruled out in all light curves using a search for correlated outliers,
providing weak constraints on the role of Oort-like comet clouds in white dwarf
pollution.
