As the real common generalisations of both orthodox transversals and adequate transversals in abundant semigroups, the concept of refined generalised quasi-adequate transversals, briefly, RGQA transversals was introduced by Kong and Wang. In this paper, for the RGQA transversal, the necessary and sufficient condition for the sets I and Λ to be bands is investigated. It is demonstrated that the sets I and Λ are both bands if and only if the RGQA transversal is weakly simplistic. Moreover, the RGQA transversal S o being weakly simplistic is different from S o being a quasi-ideal nor the abundant semigroup S satisfying the regularity condition. Finally, by means of a quasi-adequate semigroup and a band, the structure theorem for an abundant semigroup with a weakly simplistic RGQA transversal is established.