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The holographic entanglement entropy functional for higher-curvature gravities involves a weighted sum whose evaluation, beyond quadratic order, requires a complicated theory-dependent splitting of the Riemann tensor components. Using the splittings of general relativity one can obtain unambiguous formulas perturbatively valid for general higher-curvature gravities. Within this setup, we perform a novel rewriting of the functional which gets rid of the weighted sum. The formula is particularly neat for general cubic and quartic theories, and we use it to explicitly evaluate the corresponding functionals. In the case of Lovelock theories, we find that the anomaly term can be written in terms of the exponential of a differential operator. We also show that order-$n$ densities involving $n_R$ Riemann tensors (combined with $n-n_R$ Ricci's) give rise to terms with up to $2n_R-2$ extrinsic curvatures. In particular, densities built from arbitrary Ricci curvatures combined with zero or one Riemann tensors have no anomaly term in their functionals. Finally, we apply our results for cubic gravities to the evaluation of universal terms coming from various symmetric regions in general dimensions. In particular, we show that the universal function characteristic of corner regions in $d=3$ gets modified in its functional dependence on the opening angle with respect to the Einstein gravity result.