Search algorithm that returns the lowest cost path to one, of many possible goal nodes?
Yes, any of the standard search algorithms will do that.

You seem to be under the misapprehension that search algorithms stop "when they reach node&nbsp;$x$." They don't: they stop when they reach a node with any property you tell them to use. That property could be "node&nbsp;$x$", or "any node in $x_1, \dots, x_k$", or "any node with score more than&nbsp;$10$" or any other condition you choose.