What did Beth Harmon mean by "I would say that it is much easier to play chess without the burden of an Adam’s Apple."

Question

Before the final game, when the participants were interviewed by reporters, one of them asked,

"Miss Harmon, What do you say to those in the Chess Federation who accuse you of being too glamorous to be a serious chess player?"

She answered to the question saying,

"I would say that it is much easier to play chess without the burden of an Adam’s Apple."

What did she mean by that?

Answer 1

cmp's comment is essentially spot-on:

I saw it as a little dig. A battle of the sex. It’s preposterous to think that a woman can be too glamorous to play, just like it’s crazy that a man might have his voice box to worry about. A very good question, nonetheless!

Think back to when Beth was being interviewed in her room, and then later her mother reads the article while they dine in a restaurant. Beth unhappily remarks that all the article cares about is that she's a girl. They don't really talk about her chess playing, remarking that they don't even mention that she plays the Sicilian. This is a reflection of prevailing cultural attitudes of the 60's, which were wildly sexist by present day standards: the woman's place was at the home, cooking and cleaning and raising children. The article reflected that she wasn't interesting because she was a remarkable chess player, but because she was a female doing well at a "male" activity. The interest was predicated upon her gender, not so much her skills and accomplishments. She knew that male players would not have received such a treatment, and she just wanted to be treated the same, and as an equal. To be respected.

There are many other signals of the sexist attitudes and barriers that Beth works against from the very beginning all throughout the series, going all the way back to when she registered for her first tournament (and even slightly before then, when her adoptive mother rejects the notion of her working to help get some money for them; or long before then if we count the bleak lessons we see her biological mother teach her in flashbacks).

So the question from the press conference you refer to reflects that this is an ongoing problem for Beth, and that the press (or whoever holds the views the reporter inquires about) still view her through the lens of femininity. In particular, since women aren't supposed to be suited for this sort of thing, she must be struggling against some intrinsic "female" characteristic that naturally holds women back. The quality of "glamorous" here is just a dog whistle for "woman", signaling her intrinsic inadequacy.

So she wittily retorts back that it must be easier to play without an Adam's apple, which is a visible, characteristic male trait. So she inverts these presumptions by, in some sense, saying that if "being glamorous" is somehow a detriment to one's ability to play chess, then the only explanation for her success and ability is that men must suffer from some other disability that cancels out her "glamour" one.