As generalizations of Ehresmann semigroups, P -Ehresmann semigroups were firstly introduced and investigated by Jones 12. From a varietal perspective, P -Ehresmann semigroups are derived from reducts of regular ◦-semigroups. In this paper, inspired by the approach of Jones, P -Ehresmann semigroups are further extended to weakly P-Ehresmann semigroups derived instead from reducts of regular semigroups with a quasi-ideal regular ◦-transversal. After studying the basic properties of weakly P -Ehresmann semigroups, we obtain a construction theorem for these semigroups by using P -Ehresmann semigroups, left normal partial bands and right normal partial bands satisfying natural compatibility conditions. Special cases are also considered, and examples are provided to illustrate the relationships among these special cases.