We present new observations of the Kuiper belt analogues around HD 38858 and HD 20794, hosts of super-Earth mass planets within 1 au. As two of the four nearby G-type stars (with HD 69830 and 61 Vir) that form the basis of a possible correlation between low-mass planets and debris disc brightness, these systems are of particular interest. The disc around HD 38858 is well resolved with Herschel and we constrain the disc geometry and radial structure. We also present a probable James Clerk Maxwell Telescope sub-mm continuum detection of the disc and a CO J = 2–1 upper limit. The disc around HD 20794 is much fainter and appears marginally resolved with Herschel, and is constrained to be less extended than the discs around 61 Vir and HD 38858. We also set limits on the radial location of hot dust recently detected around HD 20794 with near-IR interferometry. We present High Accuracy Radial velocity Planet Searcher upper limits on unseen planets in these four systems, ruling out additional super-Earths within a fewau, and Saturn-mass planets within 10 au. We consider the disc structure in the three systems with Kuiper belt analogues (HD 69830 has only a warm dust detection), concluding that 61 Vir and HD 38858 have greater radial disc extent than HD 20794. We speculate that the greater width is related to the greater minimum planet masses (10–20 M⊕ versus 3–5 M⊕), arising from an eccentric planetesimal population analogous to the Solar system’s scattered disc. We discuss alternative scenarios and possible means to distinguish among them.