Properties of star-cluster forming regions are crucial to determining whether nascent clusters emerge of their gaseous cradle as bound entities, or whether they immediately contribute to the field star population. Thanks to the Spitzer, Herschel and Wide-Field Infrared Survey Explorer space telescopes, the characterization of cluster-forming regions has made a major leap forward. In particular, molecular clouds of the Solar neighbourhood have revealed a quadratic star formation relation. We demonstrate that such a star formation relation can be accounted for by a model in which stars form in centrally-concentrated molecular clumps with a constant star formation efficiency per free-fall time. The corresponding model consequences – (i) improved survivability of clusters after the expulsion of the residual star-forming gas, and (ii) smaller stellar age spreads in clusters formed out of higher-density gas -- are discussed. Forging ahead, we also explore the pitfalls which affect the comparison of the star-formation relation for nearby molecular clouds, at low gas surface density, with the star-formation relation for more distant compact molecular clumps, at high gas surface density.