In case of Manganese (you’ll have to do research for Phosphorus yourself), since it’s common in raw group 3, I would go rather after Ruthenium and trade down.
Use Spansh to search for bodies with vulcanism and high Ruthenium contents near your location (I’ve put Sol and range 1000Ly in this search)
or search for fungal life. It will probably be further away than volcanism but from my observations plants prefer flat areas so are easier to harvest even on high gravity planets
you might also want to add to the search to have the least percentages of other rare materials. From my experience it’s not that important to do so but YMMV
in theory you might want to search for landmarks instead of just volcanism/genuses but most of the database has data from before Odyssey so it’s not filled out much. Landmarks/POIs has changed. Whether a body has volcanism or genuses, not so much
Watch this on how to find a concentration of landmarks. You don’t need an SLF but you need to fly low for landmarks to render (below 200m).
You are looking for
Phloem Excretions on the plants. Just land in a concentration of plants and you should find some
Needle Crystals that seem to be most common on Lava Spouts. But I think I’ve found some on geysers/fumaroles in the past too
if you landed in a place and can’t find growths or protrusions you are looking for in 5 minutes, take off and try another patch of blue
If you are running EDMC with Canonn plugin, scanning a growth or geyser should put it in landmarks in Spansh for future searches
EDIT: mining will also give you raw mats but rather lower grades and I don’t know how to search for a ring so you get mats from group 3 and not another one