How can I alter the breadcrumbs on a search page?
For ease of use you might want to look at Path breadcumbs:


  This module is a solution for all problems with the breadcrumbs on
  your site!
  
  Path breadcrumbs module helps you to create breadcrumbs for any page
  with any selection rules and load any entity from the URL!


There is also Custom Breadcrumbs which was pretty awesome, but be careful if you choose this route as there is serious discussion about deprecating it in favour of Path breadcrumbs.

Hansel Breadcrumbs is useful, but I have found it INCREDIBLY complicated to get set up and working properly. I would suggest to use it only if you're going to be setting up a lot of complicated rules. If it's just for one or two pages, it's probably overkill and might end up making you hate Drupal (and no-one wants that!) ;-)