Restricted divisor functions and half Appell sums in higher-level

We examine the value distributions of coefficients in certain $q$-series related to half Appell sums in higher-level and the first moment of the Garvan's $k$-rank of partitions. We prove that these coefficients equal certain restricted divisor functions and can take any nonnegative integer value infinitely many times. As applications, we confirm a conjecture of Xiong on the coefficients of a half Lerch sum and a conjecture of Garvan and Jennings-Shaffer on the nonnegativity of spt-crank-type partitions.