  Let $g$ be a dimension function. The Generalised Baker-Schmidt Problem (1970)
concerns the $g$-dimensional Hausdorff measure ($\HH^g$-measure) of the set of
$\Psi$-approximable points on nondegenerate manifolds. The problem relates the
`size' of the set of $\Psi$-approximable points with the convergence or
divergence of a certain series. In the dual approximation setting, the
divergence case has been established by Beresnevich-Dickinson-Velani (2006) for
any nondegenerate manifold. The convergence case, however, represents a major
challenging open problem and progress thus far has been effectuated in limited
cases only. In this paper, we discuss and prove several results on the
$\HH^g$-measure on Veronese curves in any dimension $n$. As a consequence of
one of our results, we generalize recent results of Pezzoni [Acta Arith. 193
(2020), no. 3, 269-281] regarding $n=3$. This improvement evolves from a deeper
investigation on general irreducibility considerations applicable in arbitrary
dimensions. We further investigate the $\HH^g$-measure for convergence on
planar curves. We show that the monotonicity assumption on a multivariable
approximating function cannot be removed for planar curves.
