The speaker describes giving a name to their home ground, a "poor little island," and the name comes from a pill. It's a small, odd admission. Calling a place after medicine suggests something about needing to be treated, or about finding the place a bit bitter but useful. The title, "Salt cure," points the same way: salt preserves and stings, and it has long been used to heal.
In the second part, the focus moves to the shore, "rough with wear." The speaker doesn't claim the place is healing them. Instead they allow that it "may do more good than I know." That hedge is the heart of the poem. The mood is modest and quiet, with a dry humor and a hint of self-deprecation. The speaker seems unsure of their own feelings, yet open to the idea that a worn, unglamorous place could be quietly doing them good without their noticing.