On 2019 August 14, the LIGO and Virgo interferometers detected a high-significance event labelled S190814bv. Preliminary analysis of the GW data suggests that the event was likely due to the merger of a compact binary system formed by a BH and a NS. ElectromagNetic counterparts of GRAvitational wave sources at the VEry Large Telescope (ENGRAVE) collaboration members carried out an intensive multi-epoch, multi-instrument observational campaign to identify the possible optical/near infrared counterpart of the event. In addition, the ATLAS, GOTO, GRAWITA-VST, Pan-STARRS and VINROUGE projects also carried out a search on this event. Our observations allow us to place limits on the presence of any counterpart and discuss the implications for the kilonova (KN) possibly generated by this NS-BH merger, and for the strategy of future searches. Altogether, our observations allow us to exclude a KN with large ejecta mass M> 0.1Msolar to a high (>90%) confidence, and we can exclude much smaller masses in a subsample of our observations. This disfavours the tidal disruption of the neutron star during the merger. Despite the sensitive instruments involved in the campaign, given the distance of S190814bv we could not reach sufficiently deep limits to constrain a KN comparable in luminosity to AT 2017gfo on a large fraction of the localisation probability. This suggests that future (likely common) events at a few hundreds Mpc will be detected only by large facilities with both high sensitivity and large field of view. Galaxy-targeted observations can reach the needed depth over a relevant portion of the localisation probability with a smaller investment of resources, but the number of galaxies to be targeted in order to get a fairly complete coverage is large, even in the case of a localisation as good as that of this event.