For those who don't know, and for those who will know very soon, a push present is what you give your partner shortly after they've given birth. It's not customary for your girl to have to ask for it, but since women are smart and know what men don't, she may have already mentioned the topic to you. You Googled it and now here you are. 

Don't worry, dads, it's not all on you. I've written another post for moms to understand that you need help too. 

In the mix of nurses telling her WHAT to do, you telling her that whatever the heck she's doing is just stupendous will cheer her up and relieve some pressure.