The title says polynomial, so I will assume that $\beta$ is an integer. Yes, you can linearize by expanding the power and introducing a new binary variable for each resulting product of $X_i$s. For example, if $\beta=3$, one of the products is $X_1 X_2 X_3$ and you would introduce binary variable $Y_{1,2,3}$, together with constraints
\begin{align}
Y_{1,2,3} &\le X_1 \tag1\\
Y_{1,2,3} &\le X_2 \tag2\\
Y_{1,2,3} &\le X_3 \tag3\\
X_1 + X_2 + X_3 - 2 &\le Y_{1,2,3} \tag4\\
\end{align}
Constraints $(1)$ through $(3)$ enforce $Y_{1,2,3} \le X_1 X_2 X_3$.
Constraint $(4)$ enforces $Y_{1,2,3} \ge X_1 X_2 X_3$.
You can optionally relax $Y$ to be nonnegative because it will automatically take values $\{0,1\}$ when $X$ is binary.
Here's an alternative linearization for the case that $\beta$ is not necessarily integer. Suppose you know that $\sum_i a_i X_i \in S$ for some finite set $S$ that is easy to generate. For example, if the $a_i$ are positive integers, you can take $S=\{0,1,\dots,\sum_i a_i\}$. Now for each $s \in S$, let binary variable $Z_s$ indicate whether $\sum_i a_i X_i = s$. The problem is to maximize
$$\sum_{s \in S} s^\beta Z_s - \sum_{i=1}^N c_i X_i$$
subject to
\begin{align}
\sum_{s \in S} Z_s &= 1 \\
\sum_{s \in S} s Z_s &= \sum_{i=1}^N a_i X_i \tag1\\
\end{align}
If you cannot easily generate $S$ such that $\sum_i a_i X_i \in S$, you can approximate by letting $\Delta=(\sum_i a_i)/m$, taking $S=\{0,\Delta,2\Delta,\dots,\sum_i a_i\}$,
and relaxing equality constraint $(1)$ to an inequality:
$$-\frac{\Delta}{2} \le \sum_{s \in S} s Z_s - \sum_{i=1}^N a_i X_i \le \frac{\Delta}{2}$$
