There are different ways to do this. You can do that with few lines php code. Need more infos about your site (is it coustom theme or do you use cms like wordpress).
If you use wordpress, you need 2 different menus. 1 specifically menu for your homepage with the link(s). Then you can define it in functions.php for each Page. It seems that you have no experience with php-coding? In that case I recommend you to use an existing plugin like:
https://de.wordpress.org/plugins/pag...ic-menu-items/
or
https://themify.me/conditional-menus
Search for plugins "different menus for each site"