We quantify the star formation (SF) in the inner cores (R/R_200_~<0.3) of 24 massive galaxy clusters at 0.2~<z~<0.9 observed by the Herschel Lensing Survey and the Cluster Lensing and Supernova survey with Hubble. These programmes, covering the rest-frame ultraviolet to far-infrared regimes, allow us to accurately characterize stellar mass-limited (M_*_>10^10^M_{sun}_) samples of star-forming cluster members (not)-detected in the mid- and/or far-infrared. We release the catalogues with the photometry, photometric redshifts, and physical properties of these samples. We also quantify the SF displayed by comparable field samples from the Cosmic Assembly Near-infrared Deep Extragalactic Legacy Survey. We find that in intermediate-z cluster cores, the SF activity is suppressed with respect the field in terms of both the fraction (F) of star-forming galaxies (SFGs) and the rate at which they form stars (SFR and sSFR=SFR/M_*_). On average, the F of SFGs is a factor ~2 smaller in cluster cores than in the field. Furthermore, SFGs present average SFR and sSFR typically ~0.3dex smaller in the clusters than in the field along the whole redshift range probed. Our results favour long time-scale quenching physical processes as the main driver of SF suppression in the inner cores of clusters since z~0.9, with shorter time-scale processes being very likely responsible for a fraction of the missing SFG population.