Is there a variant of Dijkstra’s algorithm for partial recalculation?
Yes. A search algorithm that is able to quickly recompute the fastest path when small changes are made is known as an incremental algorithm.  For your case specifically, you'll want to use D*-Lite, the algorithm used by the Mars Rovers.

See here for more information.

(Note: D*-Lite is decently complex.  If this is for a video game, there are better ways to obtain speed-ups using off-the-shelf A* implementations)