This hole likely is part of backflow preventive, so check for rubber flapper inside.
Most of RV have the cheapest fixtures that come on the market.
I have similiar problem on my camper, where the similiar to above valves have manifold cracks. I already replaced outside shower, but my bathroom valve has faucet/shower combo with limited countertop space. I was thinking putting thermostatic valves, that I am buying from China for $75, but those are too big to fit in the corner.
You might have more space, so see if you could do the upgrade.
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