Does Ernest Hemingway appear to draw on personal experience in the way he captures the controversial themes in "A Clean, Well-Lighted Place"?

This is a dangerous question in that it assumes that the action taking place in the story is autobiographical to the author. That said, you can argue that Ernest Hemingway draws on his experiences in The Great War (World War I to most of us) to examine the nihilistic themes present in this story.

The story's climactic moment comes when the older waiter, who has been patient with the old deaf man who had tried...

This is a dangerous question in that it assumes that the action taking place in the story is autobiographical to the author. That said, you can argue that Ernest Hemingway draws on his experiences in The Great War (World War I to most of us) to examine the nihilistic themes present in this story.

The story's climactic moment comes when the older waiter, who has been patient with the old deaf man who had tried to commit suicide the previous week, is having a drink of his own and recites his version of "The Lord's Prayer." The older waiter, who we can assume either served in The Great War or was at least affected by it, replaces most of the meaningful words with "nada," which is Spanish for "nothing." Here's an example (the replaced words are in brackets): 

"Our nada [Father] who art in nada [heaven], nada [hallowed] be thy name thy kingdom nada [come] thy will be nada [done] in nada [earth] as it is in nada [heaven]..."

Hemingway, who served in The Great War, often writes about the emptiness of the war and the feelings of hopelessness that came as a result. The war is what inspired thousands of Americans, Hemingway included, to flee the United States to live as disillusioned expatriates in France drinking much of their time away. It is what led Hemingway to write The Sun Also Rises, a novel about Americans disillusioned after The Great War. It is often considered Hemingway's most autobiographical work of fiction.

These themes of disillusionment and hopelessness that Hemingway explores in "A Clean, Well-Lighted Place" definitely shares some autobiographical roots with Hemingway, but, more importantly, they explore the themes predominant in the era immediately following the war.