Sums of divisor functions and von Mangoldt convolutions in 𝔽q[T] leading to symplectic distributions

In [J. P. Keating, B. Rodgers, E. Roditty-Gershon and Z. Rudnick, Sums of divisor functions in 𝔽𝑞⁢[𝑡] and matrix integrals, Math. Z. 288 2018, 1–2, 167–198], the authors established relationships of the mean-square of sums of the divisor function 𝑑𝑘⁢(𝑓) over short intervals and over arithmetic progressions for the function field 𝔽𝑞⁢[𝑇] to certain integrals over the ensemble of unitary matrices. We consider similar problems leading to distributions over the ensemble of symplectic matrices. We also consider analogous questions involving convolutions of the von Mangoldt function.