Objective: A common challenge in model identification with clinical data is incomplete and sometimes imprecise information. In this work, we provide a method to reconstruct the corrupted input data in a clinical dataset and, jointly identify the person-specific parameters of a metabolic model describing meal-insulin-glucose-dynamics for people with type 1 diabetes (T1D). Method: The proposed method is an algorithm that iterates between nonlinear least-squares and mixed-integer quadratic programming to optimize model parameters in conjunction with sparse corrections to the input data. In order to handle long stretches of data, the optimization problem is designed to be i) computationally tractable, and ii) robust against the potential presence of significant inaccuracies corrupting a data portion. Moreover, since the pattern of the inaccuracies is specific to each person, we propose a personalized hyperparameter tuning approach. The method is applied on clinical data from 13 people with T1D. Identified model performance is compared to the performance of model identified with standard least squares (LS) method. Results: Compared to LS, identifying corrections in conjunction with model parameters on training data lead to an improvement in the model prediction capabilities on unseen data with an average 2.2% improvement in MARD for two-hour prediction horizon (p-value = 0.0006). Conclusions: The proposed method is effective in model identification for clinical data with unknown inaccuracies in the inputs. Significance: Personalized models with high accuracy can inform treatment decisions and lead to better glucose control outcomes in people with T1D.