We consider the effective theory of perturbative quantum gravity coupled to a point particle, quantizing fluctuations of both the gravitational field and the particle's position around flat space. Using a recent relational approach to construct gauge-invariant observables, we compute one-loop graviton corrections to the invariant metric perturbation, whose time-time component gives the Newtonian gravitational potential. The resulting quantum correction consists of two parts: the first stems from graviton loops and agrees with the correction derived by other methods, while the second one is sourced by the quantum fluctuations of the particle's position and energy-momentum, and may be viewed as an analog of a ``Zitterbewegung''. As a check on the computation, we also recover classical corrections which agree with the perturbative expansion of the Schwarzschild metric. Based on joint work with C. Rein and R. Verch, arXiv:2109.09753 [JHEP 01 (2022) 180].