Critical quantum metrology relies on the extreme sensitivity of a system’s eigenstates near the critical point of a quantum phase transition to Hamiltonian perturbations. This means that these eigenstates are extremely sensitive to all the parameters of the Hamiltonian. In practical settings, there always exists a degree of experimental uncertainty in the control parameters - which are approximately known quantities. Despite such uncertainties representing the most relevant source of noise in critical metrology, their impact on the attainable precision has been largely overlooked. In this work we present a general framework, interpolating between the single and multi-parameter estimation settings, allowing for the proper bookkeeping of relevant errors. We apply this framework to the paradigmatic transverse field Ising and Lipkin-Meshkov-Glick models, explicitly showing how uncertainty in control parameters impacts the sensitivity of critical sensors. For finite-size systems, we establish that there exists a trade-off between the amount of uncertainty a many-body probe can withstand while still maintaining a quantum advantage in parameter estimation.