How can I add a specific page to the 'New Tab' page?

To do this you have to make your specific site to bookmark and then drag that bookmark from library to the New Tab page on a thumbnail. You can also pin tabs in the order you want them to appear.

Mozilla support have a page where they tell you how to do it.

I just did this and it worked for me: I went to recent history and dragged the site I wanted over to an empty thumbnail spot after I deleted a bunch I didn't want. It was a lucky try after trying many other things. Hope it works for you too!