We present an augmented parton mean-field theory which (i) reproduces the $exact$ ground state, spectrum, and dynamics of the quantum spin liquid phase of Kitaev's honeycomb model; and (ii) is amenable to the inclusion of integrability breaking terms, allowing a perturbation theory from a controlled starting point. Thus, we exemplarily study dynamical spin correlations of the honeycomb Kitaev quantum spin liquid within the $K-J-\Gamma$ model which includes Heisenberg and symmetric-anisotropic (pseudo-dipolar) interactions. This allows us to trace changes of the correlations in the regime of slowly moving fluxes, where the theory captures the dominant deviations when integrability is lost. These include an asymmetric shift together with a broadening of the dominant peak in the response as a function of frequency; the generation of further-neighbour correlations and their structure in real- and spin-space; and a resulting loss of an approximate rotational symmetry of the structure factor in reciprocal space. We discuss the limitations of this approach, and also view the neutron scattering experiments on the putative proximate quantum spin liquid material, $\alpha$-RuCl$_3$, in the light of the results from this extended parton theory.