home.php is reserved for use by the Template Hierarchy as the Blog Posts Index. To create a custom template for the Site Front Page, use
Another issue: your slideshow jQuery script needs to have a no-conflict wrapper, and should not execute until the page is loaded.
Instead of this:
...you need to use this:
Aside from those two issues, I'm not sure your problem is WordPress-specific.
I notice that you're including jQuery cycle before you include jQuery itself. I'm not sure if that would cause issues or not.
From your comment:
The Theme contains...front-page.php (static page that is the front page)
I suspect that this is somehow contributing to your problem. The
front-page.php file is not a static page but rather a Theme template file. A static page is not an actual file, but merely a database entry - a post with a post-type of page. A Theme template file is a file that WordPress uses to output content based on the current context.
Please review the Codex entry for creating a static front page, and ensure you're following all instructions correctly. Once you've got your static front page configured properly, let's see if the problem persists.