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

Given a polynomial $H(x)$ over ${\mathbb{F}}_{q^n}$, we study permutation polynomials of the form $x+\gamma \mathrm{T}\mathrm{r}(H(x))$ over ${\mathbb{F}}_{q^n}$. Let $$P_H=\{\gamma \in {\mathbb{F}}_{q^n}:x+\gamma \mathrm{T}\mathrm{r}(H(x))\text{ is a permutation polynomial}\}\mathrm{.}$$ 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 $\mathrm{T}\mathrm{r}(H(x))$ 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.