use Mojolicious::Lite;
get '/render_static/:img' => sub {
    my $self = shift;
    my $img = $self->param('img');
    my $path_to_private_images = ...( $img );
    
    return $self->render_static( $path_to_private_images );
}
__DATA__
..
<img src="<%= url_for( '/render_static/yo.png' ) -%>">