  Topologically twisted $\mathcal{N} = 4$ super Yang-Mills theory has a
partition function that counts Euler numbers of instanton moduli spaces. On the
manifold $\mathbb{P}^2$ and with gauge group $\mathrm{U}(3)$ this partition
function has a holomorphic anomaly which makes it a mock modular form of depth
two. We employ the Circle Method to find a Rademacher expansion for the Fourier
coefficients of this partition function. This is the first example of the use
of Circle Method for a mock modular form of a higher depth.
