Microscopic imaging of biological samples such as pathology slides is one of the standard diagnostic methods for screening various diseases, including cancer. These biological samples are usually imaged using traditional optical microscopy tools; however, the high cost, bulkiness and limited imaging throughput of traditional microscopes partially restrict their deployment in resource-limited settings. In order to mitigate this, we previously demonstrated a cost-effective and compact lens-less on-chip microscopy platform with a wide field-of-view of >20-30 mm^2. The lens-less microscopy platform has shown its effectiveness for imaging of highly connected biological samples, such as pathology slides of various tissue samples and smears, among others. This computational holographic microscope requires a set of super-resolved holograms acquired at multiple sample-to-sensor distances, which are used as input to an iterative phase recovery algorithm and holographic reconstruction process, yielding high-resolution images of the samples in phase and amplitude channels. Here we demonstrate that in order to reconstruct clinically relevant images with high resolution and image contrast, we require less than 50% of the previously reported nominal number of holograms acquired at different sample-to-sensor distances. This is achieved by incorporating a loose sparsity constraint as part of the iterative holographic object reconstruction. We demonstrate the success of this sparsity-based computational lens-less microscopy platform by imaging pathology slides of breast cancer tissue and Papanicolaou (Pap) smears.