Let \(r\ge 2\) , \(\ell \ge 2^{r-1}+1\) and \(k\ge 2\) be integers. In this paper, we give an asymptotic formula for \(\sum _{1\le p_{1},p_{2},\ldots ,p_{\ell }\le x^{\frac{1}{r}}}\tau _{k}(p_{1}^{r}+p_{2}^{r}+\cdots +p_{\ell }^{r}),\) where \(\tau _k(n)\) represents the k-th divisor function and \(p_{1},\dots ,p_{\ell }\) are prime variables. Moreover, we also provide an asymptotic formula for \(\sum _{1\le p_{1},p_{2},\ldots ,p_{\ell }\le x^{\frac{1}{r}}}\tau _{k}(p_{1}^{r}+p_{2}^{r}+\cdots +p_{\ell }^{r})\log p_1 \log p_2 \cdots \log p_{\ell }.\) Previously, partial results of these summations were obtained by many mathematicians.
