Motivated by combinatorial applications, a complete characterization of permutation polynomials of the form \(f(x)+\gamma x\) is of significant interest. In this paper, we first study such polynomials over \(\mathbb {F}_{q^2}\) of the form \(\sum _i(x^q-x+\delta )^{s_i}+\gamma x\) , where \(s_i\) are positive integers and \(\delta , \gamma \in \mathbb {F}_{q^2}\) with \(\gamma \ne 0\) . While partial results are known for \(\gamma \in \mathbb {F}_q\) using the AGW criterion, this method fails for \(\gamma \in \mathbb {F}_{q^2}\) . To overcome this, we propose a new technique for verifying the permutation property. Using this method, we extend many known permutation polynomials of the form \(\sum _i(x^q-x+\delta )^{s_i}+x\) over \(\mathbb {F}_{q^2}\) . We further demonstrate the efficacy of our approach by generalizing some results of Liu, Jiang, and Zou, who investigated polynomials of the form \(\sum _i(x^q-x+\delta )^{s_i}+\gamma (x^q+x)\) for \(\gamma \in \mathbb {F}_{q}^*\) and \(q\in \{3^n,5^n\}\) . Finally, for a class of special-form permutation polynomials given by \(x+\gamma \textrm{Tr}_q^{q^d}(x^{q+1}+x^{2q+2})\) , where \(q=2^m\) , \(2\not \mid d\) , we present an alternative method to determine their permutation behavior.
