We quantify the star formation (SF) in the inner cores ($mathcal{R}$/$R_{200}$$leq$0.3) of 24 massive galaxy clusters at 0.2$lesssim$$z$$lesssim$0.9 observed by the $Herschel$ Lensing Survey and the Cluster Lensing and Supernova survey with $Hubble$. These programmes, covering the rest-frame ultraviolet to far-infrared regimes, allow us to accurately characterize stellar mass-limited ($mathcal{M}_{*}$$>$$10^{10}$ $M_{odot}$) samples of star-forming cluster members (not)-detected in the mid- and/or far-infrared. We release the catalogues with the photometry, photometric redshifts, and physical properties of these samples. We also quantify the SF displayed by comparable field samples from the Cosmic Assembly Near-infrared Deep Extragalactic Legacy Survey. We find that in intermediate-$z$ cluster cores, the SF activity is suppressed with respect the field in terms of both the fraction ($mathcal{F}$) of star-forming galaxies (SFG) and the rate at which they form stars ($mathcal{SFR}$ and $smathcal{SFR} = mathcal{SFR}/mathcal{M}_{*}$). On average, the $mathcal{F}$ of SFGs is a factor $sim$$2$ smaller in cluster cores than in the field. Furthermore, SFGs present average $mathcal{SFR}$ and $smathcal{SFR}$ typically $sim$0.3 dex 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$$sim$0.9, with shorter time-scale processes being very likely responsible for a fraction of the missing SFG population.
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