1) If your information belongs in OpenStreetMap, you can add it via an editor like the web editor at https://openstreetmap.org - this will also benefit all other OSM users. You can then "refresh" your Protomaps download to get a new map.
2) If there isn't many point and polygons, it may sense to add them as Leaflet layers, especially if you want them to be interactive
3) Other options are creating vector tiles of your own data and merging or displaying them in the renderer (https://github.com/mapbox/tippecanoe is a great tool to do this from GeoJSON) but I don't have much to support this yet.
I am working on making the set of information included in the tiles configurable via a JSON definition, kind of like Overpass Turbo if you've used that.
The reason I'm doing it this way is because data completeness and performance - the total size of the tiles - are direct tradeoffs. If I keep augmenting the dataset with tags people request, they'll become too big for everyone!
So I'm exploring this idea of "a la carte" tags in tile sets, but am still gathering requirements. Can you email me at brandon@protomaps.com with more details of the OSM tags you want, as well as the area you're interested in so we can work together on checking out the data availability there?
I've been using leaflet for my map tiling and it just works. The advantage of promomaps being offline is great.
So, now... how does one add their own POI's, routes, polygons??