On higher moments of Dirichlet coefficients attached to symmetric square L-functions over certain sparse sequence

Let 2 <= j <= 8 be any fixed positive integer. Let f be a normalized primitive holomorphic cusp form of even integral weight for the full modular group Gamma = SL(2, Z). Denote by lambda(sym2f) (n) the nth normalized coefficient of the Dirichlet expansion of the symmetric square L-function L(sym(2)f, s) attached to f. In this paper, we are interested in the average behaviour of the following summatory function Sigma(a2 + b2 + c2 + d2 <= x) (4)((a,b,c,d)is an element of Z) lambda(j)(sym2f)(a(2) + b(2) + c(2) + d(2)) for x >= x(0) (sufficiently large), which improves and generalizes the recent works of Sharma and Sankaranarayanan (Res Number Theory 8:19, 2022, Rend Circ Mat Palermo II Ser 72:1399-1416, 2023).