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Let $K$ be a global field of positive characteristic. We give an asymptoticformula for the number of $K$-points of bounded height on the Hilbert scheme$\text{Hilb}^2\mathbb{P}^2$ and show that by eliminating an exceptional thinset, the constant in front of the main term agrees with the prediction of Peyrein the function field setting. Moreover, we extend the analogy between theintegers and $0$-cycles on a variety $V$ over a finite field to $0$-cycles on avariety $V$ over $K$ and establish a version of the prime number theorem in thecase when $V = \mathbb{P}^2$.

Counting points on $\text{Hilb}^m\mathbb{P}^2$ over function fields