A fraction of galaxy clusters host diffuse radio sources whose origins are investigated through multi-wavelength studies of cluster samples. We investigate the presence of diffuse radio emission in a sample of seven galaxy clusters in the largely unexplored intermediate redshift range (0.3<z<0.44). In search of diffuse emission, deep radio imaging of the clusters are presented from wide band (1.1-3.1GHz), full resolution (~5-arcsec) observations with the Australia Telescope Compact Array (ATCA). The visibilities were also imaged at lower resolution after point source modelling and subtraction and after a taper was applied to achieve better sensitivity to low surface brightness diffuse radio emission. In case of non-detection of diffuse sources, we set upper limits for the radio power of injected diffuse radio sources in the field of our observations. Furthermore, we discuss the dynamical state of the observed clusters based on an X-ray morphological analysis with XMM-Newton. We detect a giant radio halo in PSZ2 G284.97-23.69 (z=0.39) and a possible diffuse source in the nearly relaxed cluster PSZ2 G262.73-40.92 (z=0.421). Our sample contains three highly disturbed massive clusters without clear traces of diffuse emission at the observed frequencies. We were able to inject modelled radio halos with low values of total flux density to set upper detection limits; however, with our high-frequency observations we cannot exclude the presence of RH in these systems because of the sensitivity of our observations in combination with the high z of the observed clusters.