WordPress Loop – Not returning projects in specific category

You’re sure that you have actual posts (project post type), that have that category selected on them, right?

Make sure the posts have an actual category selected, not a tag. Also check in the categories configuration to make sure that websites is the actual slug of the category they are assigned to.

You can also try using the actual category id (find it while editing the category in admin area, in the URL bar), using this instead:
'cat' => '1,2,3,4'

or try
'category__in' => 4

I suspect the issue is probably with incorrect slug or some other configuration

https://developer.wordpress.org/reference/classes/wp_query/#category-parameters

You can also try using a direct tax query:

$args = array(
    'post_type'     => 'project',
    'tax_query' => array(
        array(
            'taxonomy'         => 'category',
            'terms'            => 'websites',
            'field'            => 'slug',
            'include_children' => true,
        )
    )
);