We develop a new heat kernel method that is suited for a systematic study of the renormalization group flow in Hořava gravity (and in Lifshitz field theories in general). This method maintains covariance at all stages of the calculation, which is achieved by introducing a generalized Fourier transform covariant with respect to the nonrelativistic background spacetime. As a first test, we apply this method to compute the anisotropic Weyl anomaly for a (2 + 1)-dimensional scalar field theory around a z = 2 Lifshitz point and corroborate the previously found result. We then proceed to general scalar operators and evaluate their one-loop effective action. The covariant heat kernel method that we develop also directly applies to operators with spin structures in arbitrary dimensions.