Why does The Bartender have surgery marks?

Jane becomes John. John becomes a temporal agent. John get's his face burnt. John gets a new face - this is the Barkeep face.

This is why Barkeep has the scars, he is John, only with a different face.

Here is a timeline showing the same:

You can also find the full timeline explained here:

Predestination Timeline Explained

He has the surgery marks because he had surgery!

The entire point of the film is to present a series of paradoxes.

As we know from the end of the film, Ethan Hawke's character is effectively everybody. Or to put it another way, he is John, Jane, the child from the mission and the Fizzle Bomber. The wiki has some useful information, but you might find it easier to look at the short story it was based on, as it's much shorter and easier to understand.

To answer your question specifically, "Jane" falls pregnant to an older man, who then leaves her.

When trying to deliver the child, the doctors discover she is intersex. The child is kidnapped, but meanwhile the doctors realise that, for her own safety, Jane's ovaries, fallopian tubes, uterus and vagina should be removed - meaning she'll retain only male sex organs (and eventually become "John").

So the surgery marks he has are because of his past, when he was a she.

If you want more help, consider this wonderful graphic I found on reddit, showing the timeline of events:

He is an older version of Jane, the unmarried mother. In the beginning of the movie, the character's face is burned and operated on so he looks different. So he recruits his past self Jane/John to take his place.