  We study the moduli space of self-dual instantons on $\mathbb{C}P^2$. These
are described by an ADHM-like construction which allows to compute the Hilbert
series of the moduli space. The latter has been found to be blind to certain
compact directions. In this paper we probe these, finding them to correspond to
a Grassmanian, upon considering appropriate ungaugings. Moreover, the ADHM-like
construction can be embedded into a $3d$ gauge theory with a known gravity
dual. Using this, we realize in $AdS_4/CFT_3$ (part of) the instanton moduli
space providing at the same time further evidence supporting the $AdS_4/CFT_3$
duality. Moreover, upon orbifolding, we provide the ADHM-like construction of
instantons on $\mathbb{C}P^2/\mathbb{Z}_n$ as well as compute its Hilbert
series. As in the unorbifolded case, these turn out to coincide with those for
instantons on $\mathbb{C}^2/\mathbb{Z}_n$.
