On degree power sum in $P_k$-free graphs

Let $G$ be a graph on $n$ vertices with degree sequence $(d_1,d_2......d_n)$. For a real $p \geq 1$, let $D_p(G)=\sum_{i=1}^nd_i^p$. A Tur\'an-type problem of degree power sum was initiated by Caro and Yuster \cite{caro2000degpower}: determining the function D_p(n,H) :=\max \{D_p(G): \text{G$is an$n$-vertex$H-free graph}\}. They obtained some exact values for certain graphs $H$. For a path $P_k$, they mentioned that ``a close examination of the proof of Theorem 1.2 shows that the value of $n_0(k)$ in the statement of the theorem is $O(k^2)$", namely, they could show the $n$-vertex $P_k$-free graph with maximum degree power sum is $W_{n,k-1,\lfloor \frac{k}{2} \rfloor -1} = K_{\lfloor \frac{k}{2} \rfloor -1} \vee \left((n - \lceil \frac{k}{2} \rceil)K_1 \cup K_{1+k-2\lfloor \frac{k}{2} \rfloor} \right)$ when $n \geq c k^2$ for some constant $c$. In this note, we improve their result to a linear size of $k$ by a different approach. The bound is tight up to a constant factor.