  We prove weak uniqueness of mild solutions for general classes of SPDEs on a
Hilbert space. The main novelty is that the drift is only defined on a
Sobolev-type subspace and no H\"older-continuity assumptions are required. This
framework turns out to be effective to achieve novel uniqueness results for
several specific examples. Such wide range of applications is obtained by
exploiting either coloured or rougher-than-cylindrical noises.
