  Let $\gcd(j,k)$ be the greatest common divisor of the integers $j$ and $k$.
In this paper, we give several interesting asymptotic formulas for weighted
averages of the $\gcd$-sum function $f(\gcd(j,k)) $ and the function
$\sum_{d|k, d^{s}|j}(f*\mu)(d) $ for any positive integers $j$ and $k$, namely
$$ \sum_{k\leq x}\frac{1}{k^{r+1}}\sum_{j=1}^{k}j^{r}f(\gcd(j,k)) \quad
\text{and} \quad \sum_{k\leq x}\frac{1}{k^{s(r+1)}}\sum_{j=1}^{k^s}j^{r}
\sum_{\substack{d|k d^{s}|j}}(f*\mu)(d), $$ with any fixed integer $s> 1$ and
any arithmetical function $f$. We also establish mean value formulas for the
error terms of asymptotic formulas for partial sums of $\gcd$-sum functions
$f(\gcd(j,k)). $
