We present a submillimetre survey of seven high-redshift galaxy clusters (0.64 < z < 1.0) using the Submillimetre Common-User Bolometer Array (SCUBA) at 850 and 450 m. The targets, of similar richness and redshift, are selected from the Red-sequence Cluster Survey (RCS). We use this sample to investigate the apparent excess of submillimetre source counts in the direction of cluster fields compared to blank fields, as seen in the literature. The sample consists of three galaxy clusters that exhibit multiple optical arcs due to strong gravitational lensing, and a control group of four clusters with no apparent strong lensing. A tentative excess of 2.7σ is seen in the number density of submillimetre sources within the lensing cluster fields compared to that in the control group. Ancillary observations at radio, mid-infrared, optical and X-ray wavelengths allow for the identification of counterparts to many of the submillimetre luminous galaxies (SMGs), and provide improved astrometry and redshift constraints. Utilizing photometric redshifts, we conclude that at least three of the galaxies within the lensing fields have redshifts consistent with the clusters and implied infrared luminosities of ∼1012 L⊙. The existence of submillimetre cluster members may therefore be boosting source counts in the lensing cluster fields, which might be an effect of the dynamical state of those clusters. However, we find that the removal of potential cluster members from the counts analysis does not entirely eliminate the difference between the cluster samples. We also investigate possible occurrences of lensing between background submillimetre sources and lower redshift optical galaxies, though further observations are required to make any conclusive claims. Although the excess counts between the two cluster samples have not been unambiguously accounted for, these results warrant caution for interpreting submillimetre source counts in cluster fields and point source contamination for Sunyaev–Zel’dovich surveys.