what letter are you talking about?
OK, out of the bath with Mrs. Cusser, maybe had too much wine.
Turned on captions and re-watched that part. Deborah at their date had told Noodles that he "was the only person I've ever cared about", and of course later in the "date" Noodles got mixed signals and raped Deborah. After that, in the added footage at a speakeasy Noodles (drunk) tells the newly-rented Eve ($500, instead of her usual $500 fee) that he wanted to call her "Deborah", and she responded "I've done more complicated things for less".
And during their time together in bed, Noodles mumbles the name "Deborah" a few times, so Eve left him that note, signed as "Deborah". The note stated "So long and Thanks. Next time less money and more work, I hope. Tel. PL2-5630 Deborah". So Eve was OK with Noodles calling her "Deborah", and wanted Noodles to call her again. So had nothing to do with the real Deborah, except that was who Noodles was thinking of. Next scene Noodles and Deborah are both drinking, and Noodles goes to the station to watch Deborah depart. And I believe Deborah sees him and then pulls down the shade, to go on with her new life.
Drunk Noodles does mumble during that scene, why it's not so easy to decipher.