  We study the asymptotic behavior of Hitchin's hyperk\"ahler metric on the
moduli space of rank two irregular Higgs bundles over $\mathbb{C}P^1$. Along a
generic curve, we prove that the Hitchin metric is asymptotic to the semiflat
metric at an arbitrary polynomial order. When there are no weakly parabolic
singularities, the rate is exponential. In the case of four-dimensional moduli
spaces, we prove that the semiflat metric is asymptotic to an ALG/ALG$^\ast$
model metric.
