In this paper, we study the small prime solutions of equation \(a_1p_1+a_2p_2+a_3p_3^k=n\) , where \(a_1,a_2,a_3\) are non-zero integers satisfying \((a_i,a_j)=1, 1\le i<j\le 3\) , and \(k\ge 4, n\) are integers. Let \(\sigma _k^{-1}=\min \{2^{k-1},k(k-1)\}\) . For any \(\varepsilon >0\) , we establish (i) if \(a_1,a_2,a_3\) are all positive, and \(n\gg \max \{2,|a_1|,|a_2|,|a_3|\}^{3k\sigma _k^{-1}+1+\varepsilon }\) , then the above equation is solvable in primes \(p_j\) , and (ii) if \(a_1,a_2,a_3\) are not all of the same sign, then the above equation has prime solutions satisfying \(\max \{ p_1,p_2,p_3^k \}\ll |n|+\max \{2,|a_1|,|a_2|,|a_3|\}^{3k\sigma _k^{-1}+\varepsilon }\) , where the implied constants depend only on \(\varepsilon \) . This is a quantitative result compared with the qualitative result of Ming-Chit Liu and Kai-Man Tsang.
