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We analyse TESS photometry of DMPP-1 (HD 38677; TIC 66560666), a nearby F8V star hosting hot super-Earth planets and a warm Neptune. Using the Transit Least Squares algorithm and other methods we find a transit signal at $P=3.2854^{+0.0032}_{-0.0025}$ d with depth 87$^{+25}_{-30}$ ppm and false alarm probability 1.6%. This is shallower than hitherto published TESS discoveries. The 3.285 d signal is recovered for several, but not all, methods for detrending stellar astrophysical variability. Further observations are needed to improve the significance of the detection. If this transit were due to an Earth-like rocky planet it would have been detected in the RV data, but it is not. The TESS data cover seven individual transits, one of which is consistent with zero depth. The insolation of the putative planet is $990 {\rm S_{\oplus}}$, typical of fluxes experienced by the three known catastrophically disintegrating exoplanets (CDEs). The transits can be self-consistently attributed to a CDE with a mass below the RV detection threshold. We searched for transits of the known RV planets, finding null results and detection thresholds of $< 100$ ppm, which we quantify for each. The DMPP-1 planetary system was discovered as a consequence of circumstellar gas attributed to ablation of hot planets. The RV planets may have been ablated to near-pure iron cores. We place limits on the orbital inclinations of the RV planets where the expected transit depth exceeds the detection threshold. If the 3.2854 day transit detection is confirmed, e.g. with CHEOPS photometry, DMPP-1 would be a first-rate target for JWST spectroscopy.