Permutation polynomials of the form $x+\gamma \mathrm{Tr}(H(x))$

Given a polynomial $H(x)$ over $\mathbb{F}_{q^n}$, we study permutation polynomials of the form $x + \gamma \mathrm{Tr}(H(x))$ over $\mathbb{F}_{q^n}$. Let $$P_H=\{\gamma\in \mathbb{F}_{q^n} : x+\gamma \mathrm{Tr}(H(x))~\text{is a permutation polynomial}\}.$$ We present some properties of the set $P_H$, particularly its relationship with linear translators. Moreover, we obtain an effective upper bound for the cardinality of the set $P_H$ and show that the upper bound can reach up to $q^n - q^{n - 1}$. Furthermore, we prove that when the cardinality of the set $P_H$ reaches this upper bound, the function $\mathrm{Tr}(H(x))$ must be an $\mathbb{F}_q$-linear function. Finally, we study two classes of functions $H(x)$ over $\mathbb{F}_{q^2}$ and determine the corresponding sets $P_H$. The sizes of these sets $P_H$ are all relatively small, even only including the trivial case.