The aim of this paper is to study further characterizations and the relationships of $\delta p$-normal spaces, almost $\delta p$-normal spaces and mildly $\delta p$-normal spaces. We introduce the notion of $g\delta pr$-closed sets. Also, we obtain properties of $g\delta pr$-closed sets and the relationships between $g\delta pr$-closed sets and the related generalized closed sets. By using $g\delta pr$-closed sets, we introduce new forms of generalized $\delta$-precontinuity. Moreover, we obtain new characterizations of $\delta p$-normal spaces, almost $\delta p$-normal spaces and mildly $\delta p$-normal spaces and preservation theorems.