In linear least squares the parameter estimates are: $\hat{\beta} = \left(X^{\top}X\right)^{-1}X^{\top}y$. In Ridge regression the standardized parameter estimates are given by $\hat{\beta}_{\Gamma} = \left(X^{\top}X + \Gamma\right)^{-1}X^{\top}y$.
Variance covariance estimators use leverage values to account for bias in the variance covariance estimates. For linear least squares a few examples include HC2 and HC3 which have effective residuals as: $\tilde{u} = \frac{\hat{u}}{1 - h_{ii}}$ and $\tilde{u} = \frac{\hat{u}}{\left(1 - h_{ii}\right)^{2}}$. $h_{ii}$ are the diagonal values of the projection matrix of the model matrix $h = X\left(X^{\top}X\right)^{-1}X^{\top}$.
My question is: for Ridge Regression, would the leverage adjusted residuals use $h = X\left(X^{\top}X\right)^{-1}X^{\top}$ or do they use a different for due to the fact that
$\hat{y}_{OLS} = X\left(X^{\top}X\right)^{-1}X^{\top}y \ne \left(X^{\top}X + \Gamma\right)^{-1}X^{\top}y = \hat{y}_{\Gamma}$? If different what is the form? How does it generalizes to clusters?