Moment formulas of Siegel transforms with congruence conditions in dimension 2

We compute the first and second moment formulas for Siegel transforms related to problems counting primitive lattice points in the real plane with congruence conditions. As applications, we derive an analog of Schmidt's random counting theorem and the quantitative Khintchine theorem for irrational numbers, approximated by rational numbers $p/q$, where we place a congruence-conditional constraint on the vector $(p,q)$.