This paper examines the syntax of dative possessors found with existential unaccusative verbs in Polish, called lack-type verbs. These verbs cannot be viewed as dyadic unaccusatives in Polish, since the dative possessor they contain behaves like an external argument, because it can bind subject-oriented anaphors. The dative possessor differs from items introduced in the specifier of a high or low ApplP, which can never bind anaphors in Polish. The unaccusativity of lack-type verbs and the external argument status of the dative possessor are reconciled by merging the dative possessor in the specifier of VoiceP, the position typical of external arguments, which is nonetheless non-thematic (expletive). Adopting Myler’s (2016; 2018) predicative possession, we argue that the possessor role of the dative DP does not come from the lack-type verb itself but is mediated by the possessed DP, denoting the relation between the possessor and the possessee. The dative does not merge within the possessed DP, but it saturates the possessor theta role once it is merged in the specifier of expletive VoiceP via Delayed Gratification.