This paper presents a dedicated search for exotic decays of the Higgs boson to a pair of new spin-zero particles, [Formula: see text], where the particle a decays to b-quarks and has a mass in the range of 20-60 GeV. The search is performed in events where the Higgs boson is produced in association with a [Formula: see text] boson, giving rise to a signature of a lepton (electron or muon), missing transverse momentum, and multiple jets from b-quark decays. The analysis is based on the full dataset of pp collisions at [Formula: see text] recorded in 2015 by the ATLAS detector at the CERN Large Hadron Collider, corresponding to an integrated luminosity of 3.2 [Formula: see text]. No significant excess of events above the Standard Model prediction is observed, and a [Formula: see text] confidence-level upper limit is derived for the product of the production cross section for [Formula: see text] times the branching ratio for the decay [Formula: see text]. The upper limit ranges from 6.2 pb for an a-boson mass [Formula: see text] to 1.5 pb for [Formula: see text].