Partially recorded data are frequently encountered in many applications. In practice, such datasets are usually clustered by removing incomplete cases or features with missing values, or by imputing missing values, followed by application of a clustering algorithm to the resulting altered data set. Here, we develop clustering methodology through a model-based approach using the marginal density for the observed values, using a finite mixture model of multivariate $t$ distributions. We compare our algorithm to the corresponding full expectation-maximization (EM) approach that considers the missing values in the incomplete data set and makes a missing at random (MAR) assumption, as well as case deletion and imputation. Since only the observed values are utilized, our approach is computationally more efficient than imputation or full EM. Simulation studies demonstrate that our approach has favorable recovery of the true cluster partition compared to case deletion and imputation under various missingness mechanisms, and is more robust to extreme MAR violations than the full EM approach since it does not use the observed values to inform those that are missing. Our methodology is demonstrated on a problem of clustering gamma-ray bursts and is implemented in the https://github.com/emilygoren/MixtClust R package.