For a finite group \(G\), let \(\psi(G)\) denote the sum of element orders of \(G\). This function was introduced by Amiri, Amiri, and Isaacs in 2009 and they proved that for any finite group \(G\) of order \(n\), \(\psi(G)\) is maximum if and only if \(G \simeq \mathbb{Z}_n\) where \(\mathbb{Z}_n\) denotes the cyclic group of order \(n\). Furthermore, Herzog, Longobardi, and Maj in 2018 proved that if \(G\) is non-cyclic, \(\psi(G) \leq \frac{7}{11} \psi(\mathbb{Z}_n)\). Amiri and Amiri in 2014 introduced the function \(\psi_k(G)\) which is defined as the sum of the \(k\)-th powers of element orders of \(G\) and they showed that for every positive integer \(k\), \(\psi_k(G)\) is also maximum if and only if \(G\) is cyclic. In this paper, we have been able to prove that if \(G\) is a non-cyclic group of order \(n\), then \(\psi_k(G) \leq \frac{1+3.2^k}{1+2.4^k+2^k} \psi_k(\mathbb{Z}_n)\). Setting \(k=1\) in our result, we immediately get the result of Herzog et al. as a simple corollary. Besides, a recursive formula for \(\psi_k(G)\) is also obtained for finite abelian \(p\)-groups \(G\), using which one can explicitly find out the exact value of \(\psi_k(G)\) for finite abelian groups \(G\).