A positive integer \(n\) is square-full if for any prime divisor \(p\) of \(n\), square of \(p\) divides \(n\). Let \(f(n)\) be the characteristic function of the set of square-full integers, and \[ s_q(n)=\sum_{d|\gcd(n,q)}d\,f\left(\frac{q}{d}\right). \] In this paper, the author approximates the double sum \[ S(x,y)=\sum_{n\leq y}\sum_{q\leq x}s_q(n), \] for the real numbers \(x\) and \(y\) satisfying \(x^{4/3}\log x\ll y\ll x^{14/9}\log^{-4}x\). The proof runs over standard complex integration methods, based on Perron’s formula.