We measure the 850-$mu$m source densities of 46 candidate protoclusters selected from the Planck High-z catalogue (PHz) and the Planck Catalogue of Compact Sources (PCCS) that were followed up with Herschel-SPIRE and SCUBA-2. This paper aims to search for overdensities of 850-$mu$m sources in order to select the fields that are most likely to be genuine protoclusters. Of the 46 candidate protoclusters, 25 have significant overdensities ($>$5 times the field counts), 11 have intermediate overdensities (3--5 times the field counts) and 10 have no overdensity ($<$3 times the field counts) of 850-$mu$m sources. We find that the enhanced number densities are unlikely to be the result of sample variance. Compared with the number counts of another sample selected from Plancks compact source catalogues, this [PHz+PCCS]-selected sample has a higher fraction of candidate protoclusters with significant overdensities, though both samples show overdensities of 850-$mu$m sources above intermediate level. Based on the estimated star-formation rate densities (SFRDs), we suggest that both samples can efficiently select protoclusters with starbursting galaxies near the redshift at which the global field SFRD peaks ($2 < z < 3$). Based on the confirmation of overdensities found here, future follow-up observations on other PHz targets may greatly increase the number of genuine DSFG-rich clusters/protoclusters.
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