I don't have an opinion on the handwritten item, other than to say I've seen far stranger things.

From what I've read, it was brought to market 5 or 6 years ago by uber-collector and former DiMaggio business manager Bert Padell. This is its provenance according to an old Padell interview in Sports Collectors' Digest:

“I have a letter from the president of the Boston Red Sox in January of 1918, his next-to-the-last last year as a Red Sox (and where Ruth pitched extensively) that said, ‘If you win 24 games, you get a $1,000 bonus (he had won 23 and 24 games in the previous two seasons), and if you win 30 games, you get a $1,500 bonus.’ “And it’s signed George Herman ‘Babe’ Ruth, and by the president of the Red Sox, Harry Frazee. A man in Westchester, N.Y., that I met through one of my clients in the entertainment business had this big box of items from the Red Sox in his attic, and it had been there for 30 years or more. And there was all this memorabilia, and he asked me if I wanted to buy it.” Padell told him that he had never bought anything in his life. So he brought the stuff to Padell’s Midtown Manhattan office to have him look at it, and there are the letter and two contracts signed by Ruth, and he also had a lot of Red Sox contracts. “He offered them all to me for $5,000,” said Padell. “And I told him again that I had never bought anything, and he said he needed the money and he was in debt. So I said, OK. This was 25 years ago or so. And I paid him.” And then Padell got a call from him the next day, and he said that a friend of his had called and offered him $25,000 for two of the Ruth items. “And I asked who his friend was, and he said, ‘Barry Halper.’ And I said, ‘You’ve got to be kidding.’ “And he let me photocopy the Ruth contracts and I got the letter and I got the Red Sox players contracts.”

I don't know if that would be considered solid provenance, but Padell acquired the item about 30 years ago, the previous owner supposedly had it for 30+ years, and perhaps more importantly it was offered to Padell with a number of similar vintage contracts and related documents whose authenticity have apparently not (yet) been questioned.