Conventional Bayesian inference typically assumes that the computational model replicates the true mechanism behind data generation. As a result, calibrated model parameters are often biased, leading to deficient predictive skills. Augmenting model outputs with statistical correction terms may remove the predictive bias, but it can violate physical laws, make the calibrated model ineffective for predicting non-observable quantities, and experience identifiability challenges in distinguishing between data noise and model error. This work will present a framework for representing, quantifying and propagating uncertainties due to model structural errors by embedding stochastic correction terms in the model. The embedded correction approach ensures physical constraints are satisfied, and renders calibrated model predictions meaningful and robust with respect to structural errors. The physical inputs and correction parameters are simultaneously inferred via surrogate-enabled Markov chain Monte Carlo. With a polynomial chaos characterization of the correction term, the approach allows efficient decomposition of uncertainty that includes contributions from data noise, parameter posterior uncertainty, and model error. The developed structural error quantification workflow is implemented in UQ Toolkit (www.sandia.gov/uqtoolkit). We will discuss the challenges associated with the increased dimensionality of the associated Bayesian problem, and steps that are taken to alleviate them.