Using (r, s)-preopen sets [14] and pre-${omega}_t$-closures [6], a new kind of covering property $P^t_{({omega}_r,s)}$-closedness is introduced in a bitopological space and several characterizations via filter bases, nets and grills [30] along with various properties of such concept are investigated. Two new types of cluster sets, namely pre-(${omega}_r$, s)t-${ heta}_f$-cluster sets and (r, s)t-${ heta}_f$-precluster sets of functions and multifunctions between two bitopological spaces are introduced. Several properties of pre-(${omega}_r$, s)t-${ heta}_f$-cluster sets are investigated and using the degeneracy of such cluster sets, some new characterizations of some separation axioms in topological spaces or in bitopological spaces are obtained. A sufficient condition for $P^t_{({omega}_r,s)}$-closedness has also been established in terms of pre-(${omega}_r$, s)t-${ heta}_f$-cluster sets.