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In this paper, we establish an asymptotic formula for the sum \\begin{equation*}\n  \\mathop{\\sum}_{\\substack{1 \\leqslant n_1, n_2 \\leqslant X^{1/2} \\\\ 1 \\leqslant n_3 \\leqslant X^{1/k}}} A(Q(n_1,n_2) + n_3^k)\\mathsf{a}(n_3), \\end{equation*} where $\\mathsf{a}(n)$ is either von-Mangoldt function or identity function, and $Q(x,y) \\in \\mathbb{Z}[x,y]$ is a binary quadratic polynomial. 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Let $k \\geqslant3$ be an integer. In this paper, we establish an asymptotic formula for the sum \\begin{equation*}\n  \\mathop{\\sum}_{\\substack{1 \\leqslant n_1, n_2 \\leqslant X^{1/2} \\\\ 1 \\leqslant n_3 \\leqslant X^{1/k}}} A(Q(n_1,n_2) + n_3^k)\\mathsf{a}(n_3), \\end{equation*} where $\\mathsf{a}(n)$ is either von-Mangoldt function or identity function, and $Q(x,y) \\in \\mathbb{Z}[x,y]$ is a binary quadratic polynomial. 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