We investigate finite commutative groupoids $\mathcal{G}=\langle G,\circ\rangle$ such that $g\circ h\neq g$ for all elements $g,h$ of $\mathcal{G}$. First, we show that for any such groupoid, its weak (i.e. partial) subgroupoid lattice uniquely determines its subgroupoid lattice. Next, we characterize the lattice of all weak subgroupoids of such a groupoid. This is a distributive finite lattice satisfying some combinatorial conditions concerning its atoms and join--irreducible elements.