NGC 5846 is a giant elliptical galaxy with a previously well studied globular cluster system (GCS), known to have a bimodal colour distribution with a remarkably high red fraction. Here we revisit the central galaxy regions searching for new globular cluster (GC) candidates, and measure magnitudes, colours and sizes for them. We also search for their X-ray counterparts. We use archival Hubble Space Telescope WFPC2 images, from which we modelled and subtracted the host light distribution to increase the available sample of GCs. We performed photometry on the central objects, and measured sizes and coordinates for the entire GC system known in this galaxy. We detect two dozen previously unknown GC candidates in the central regions. Reliable sizes are obtained for about 60 GCs; their typical effective radii are in the range 3-5 pc. The largest clusters are located in the central regions. We find 7 X-ray counterparts to GCs, most of them in the central region. They are among the most luminous X-ray sources in NGC 5846. They are also optically luminous, compact and belong to the red subpopulation. We conclude that the GCS of NGC 5846 is composed of relatively luminous X-ray sources.