Yes you can eat a burger with a little pink in it.
The burger with a little pink in it is safe to eat as long as you've cooked the burger to an internal temperature of 160 F throughout the burger.
If the burger has a lot of pink to it and not just a little bit then you need to cook the burger longer to remove most if not all of the pink.
Eating burgers with even a little pink is not very appetizing in my experience so I like my burgers to be cooked all the way through and not have a trace of pink.
But if it's just a tiny bit of pink to the burger then it's safe to eat but may not be very good to eat.