You basically have two options:
Use PATCH
(but note that you have to define your own media type that specifies what will happen exactly)
Use POST
to a sub resource and return 303 See Other with the Location header pointing to the main resource. The intention of the 303 is to tell the client: "I have performed your POST and the effect was that some other resource was updated. See Location header for which resource that was." POST/303 is intended for iterative additions to a resources to build up the state of some main resource and it is a perfect fit for partial updates.