How are you creating the image to print? If you’re doing it via browser screenshot or “share .png” or existing map tiles then you’re limited to what’s on those map tiles in the first place. Different tile sources have different levels of information - you might find that one of the ones at http://bigmap.osmz.ru/ works better for your needs (presumably you want street names but almost nothing else).
Something else that might help you is https://umap.openstreetmap.fr/en/ - that’d allow you to annotate a map.
Beyond that you could create your own map tiles, but that’s a bit more work.
