WP_QUERY Get posts by category and similar name (Like)

Revisited and simplified answer:

You can try:

$args = [
    'post_type'     => 'post',
    'post_status'   => 'publish',
    'category_name' => 'projects',
    '_name__like'   => 'proj*'         // <-- our new input argument!
];
$my_query = new WP_Query( $args );

where we’ve created the _name__like input argument. It supports wildcard *, for example:

    '_name__like'    => 'a*b*'        

Note that draft posts don’t have post_name set before they’re published.

We use the following plugin to support this new argument:

/**
 * Plugin Name: Support for post name like in WP_Query
 * Description: Uses the _name__like argument and supports wildcard *.
 * Plugin URI:  http://wordpress.stackexchange.com/a/136758/26350
 * Author:      Birgir Erlendsson (birgire)
 * Version:     0.0.1
 */

add_filter( 'posts_where', function( $where, $q )
{
    if( $name__like = $q->get( '_name__like' ) )
    {
        global $wpdb;
        $where .= $wpdb->prepare(
            " AND {$wpdb->posts}.post_name LIKE %s ",
            str_replace( 
                array( '**', '*' ), 
                array( '*',  '%' ),  
                mb_strtolower( $wpdb->esc_like( $name__like ) ) 
            )
        );
    }       
    return $where;
}, 10, 2 );