We consider a class of nonstandard high-order PDEs models, based on the ($p(·)$, $q(·)$)-Kirchhoff operator with variable exponents for the image denoising problem. We theoretically analyse the proposed non-linear model. Then, we use linearization method based on a fixed-point iterative technique and we also prove the convergence of the iterative process. The model has a multiscale character which follows from an adaptive selection of the exponents $p(·)$ and $q(·)$. The latter task helps to capture, highlight and correlate major features in the images and optimize the smoothing effect. We use Morley finite-elements for the numerical resolution of the proposed model and we give several numerical examples and comparisons with different methods.