Summary: Let \(\phi(z)\) be a primitive Hecke-Maass cusp forms with Laplace eigenvalue \(\frac{1}{4}+t^2\). Denote by \(L(s,\mathrm{sym}^m\phi)\) the \(m\)-th symmetric power \(L\)-function associated to \(\phi\) and by \(\lambda_{\mathrm{sym}^m\phi}(n)\) the \(n\)-th coefficient of the Dirichlet expansion of \(L(s,\mathrm{sym}^m\phi)\). For any nonzero integer \(\ell\) we prove \[ \sum\limits_{n\leqslant x}|\lambda_{\phi}(n)\lambda_{\phi}(n+\ell)|\ll_{\phi,\ell}\frac{x}{(\log x)^{0.187}}\quad (x\geqslant 3). \] This improves R. Holowinsky’s corresponding result [Ann. Math. (2) 172, No. 2, 1499–1516 (2010; Zbl 1214.11054)], which requires \(\tfrac{1}{6}\) in place of 0.187. for all \(x\geqslant 3\). Further assuming that \(L(s,\mathrm{sym}^{10}\phi)\) and \(L(s,\mathrm{sym}^{12}\phi)\) are automorphic cuspidal, we obtain a conditional generalization to the symmetric square case: \[ \sum\limits_{n\leqslant x}|\lambda_{\mathrm{sym}^2\phi}(n)\lambda_{\mathrm{sym}^2\phi}(n+\ell)|\ll_{\phi,\ell}\frac{x}{(\log x)^{0.196}}\quad (x\geqslant 3). \]