Via numerical computations, we study plane Poiseuille flow of a viscoplastic fluid in channels with the presence of a superhydrophobic (SH) groovy lower wall, where air is trapped inside the SH wall cavities. We assume this liquid/air interface to be flat while it is pinned at the groove edges. Our main focus is on the thin channel limit, for which the channel height is typically smaller than the groove period. To quantify the viscoplastic rheological behaviour, we rely on the Bingham model, via the Papanastasiou regularization method. We consider a no-slip boundary condition at the liquid/solid contacts while we model the liquid/air interface boundary using the Navier slip law. Focusing on the creeping flow limit (i.e. the Reynolds number is fixed to R=0.01), our main flow parameters are the Bingham number (B), the slip number (b), the groove periodicity length (ℓ), and the slip area fraction (φ). Our computational results quantify the effects of these parameters on the flow velocity fields, the unyielded plug zones and their areas, the effective slip lengths, and the friction factors. The shape of the unyielded plug zones are captured, allowing us to classify our flows into four main regimes, based on the characteristics of the unyielded zones, in particular the presence/absence of a broken center plug and/or an SH wall plug. We also find that the effective slip length generally increases with an increase in b, converging to a profile once an SH wall plug appears. Finally, the friction factor decreases with an increase in b, especially at larger B.