  In this paper, we first show that Koll\'ar's injectivity theorem on
cohomologies holds for projective generalised pairs with mild singularities
when the nef part of the generalised pair is b-abundant. Moreover, by applying
the methods from complex analytic geometry, we show that the injectivity
theorem holds for generalised Kawamata log terminal (g-klt) generalised pairs
on projective surfaces without any extra conditions on the nef parts of the
generalised pairs.
