We measure the 850-μm 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-μm 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-μm 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 Planck's 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-μm 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 dusty star-forming galaxy-rich clusters/protoclusters.