Of American-English origin, the phrase Mr Fixit (also Mr Fix It, Mr Fix-It, etc.) designates a man who fixes something, especially a man who, often illicitly, arranges matters or sets up deals.

The meaning of this phrase is similar to that of the noun fixer, denoting one who, often illicitly, arranges or adjusts matters—as in the following passage from The Confessions of a Con Man as Told to Will Irwin (New York: B. W. Huebsch, 1909), by the U.S. author Will Irwin (1873-1948):

I Join the Circus and Elope With Minnie, the Elephant

I guess what I have to say about circuses will amount to exposing the show business. People in general know very little about it. They suppose that the profits come from the ticket wagon, with perhaps a little extra for short change, peanuts and pink lemonade. As a matter of fact, eight out of every ten dollars of profit have come in the past from confidence outfits and crooked gambling games, which follow the show and are as much a part of its business as the elephants. […]
As a general rule, the smaller the circus the more corrupt it is in this respect. Many of the little ones have been run by confidence men simply as blinds for skin games. In my time there was a regular system of profit-sharing between the gamblers and the show. At the head of the outfit stood the “fixer,” whose job it was to bribe or stall city officials so that the gamblers could proceed with reasonable security, and to square it with the suckers.

These are the earliest occurrences of the phrase Mr Fixit that I have found, in chronological order:

1-: From The Arizona Republican (Phoenix, Arizona) of Thursday 15^th November 1906:

IT IS PERFECT.

The territorial fair this year is the first function ever brought to our notice with which everybody was pleased. There is no dissenting voice. Even the professional Mr. Fixit (we have no one person in mind; there are always some of the family in every community) has not come forward to suggest that the fair might have been better if it had been a little different.

2-: From The San Francisco Call (San Francisco, California) of Friday 29^th March 1907:

Washington, March 28.—[…] It was learned today that Senator Murray Crane ¹ of Massachusetts, who is known here as the original “Mr. Fix It,” and who is always interposing to straighten out a tangle, has been meddling in the Ohio situation. He made several suggestions of compromise […].

¹ Winthrop Murray Crane (1853-1920) was a Senator from Massachusetts from 1904 to 1913.

3-: From Senate Is Full of Opponents of Roosevelt ², by Ira E. Bennett, published in The San Francisco Call (San Francisco, California) of Sunday 14^th April 1907:

Washington, April 13.—[…] Roosevelt has a stalwart friend, of course, in Henry Cabot Lodge ³, but if he counts too much on Lodge’s soft-spoken colleague, Murray Crane, he may be disappointed. Crane is Boss Aldrich’s ⁴ understudy in the Senate, and is rapidly becoming a power. He is known widely as “Mr. Fixit,” on account of his remarkable powers of compromise and conciliation.

² Theodore Roosevelt (1858-1919) was the 26^th President of the USA from 1901 to 1919.
³ Henry Cabot Lodge (1850-1924) was a Senator from Massachusetts from 1893 to 1924.
⁴ Nelson Wilmarth Aldrich (1841-1915) was a Senator from Rhode Island from 1881 to 1911.

3-: From The San Francisco Call (San Francisco, California) of Monday 9^th March 1908, about the presidential campaign:

Senator Crane [is] fighting for his political life. Crane has sought to avoid this battle and he may yet succeed in dodging the unpleasantness. He has little stomach for the fight. He likes the easier way and they call him Mr. Fixit.

4-: From Shorty Does a Parlor Stunt, a short story by the U.S. author Sewell Ford (1868-1946), published in several newspapers on Sunday 5^th April 1908—for example in The Rocky Mountain News (Denver, Colorado):

It was kind of entertainin’, knowin’ the inside details, to see them two couple dodgin’ each other; Reggie towin’ Gussie off one way, and Marjorie leadin’ the Widow Bill off the other, while everybody else was in cahoots to get ’em together and give ’em a chance to make up.
“Looks like your little reunion program was hung on the hooks; eh, Sadie?” says I.
“There, Shorty, don’t rub it in!” says she. “And it’s too silly of them, acting this way! Really, they think the world of each other; but Madge is just pig headed enough to keep it up, and maybe run off with that frowzy little Count the minute she gets back to town. If she does, Reggie is going to take it mighty hard. Can’t you think of some way to break up that absurd combination?”
“What! with my cabbage head?” says I.
“Modesty, my boy, modesty!” says Pinckney, driftin’ up just then. “We all know, Shorty, that when it comes to using social finesse, you can give us—”
“Ah, stow it in the tin!” says I. “You can’t pin any josh medals on me.”
But when Pinckney gets to runnin’ a joke that he’s thought up all by himself he works it overtime. So he goes around whisperin’ confidential that it’s all right now, since I’ve been put on the job. He gets away with it too; for at dinner I see ’em pointin’ me out as Mr. Fixit, the broken vow specialist, called in for this one occasion.

5-: From The Theatres, published in the Perth Amboy Evening News (Perth Amboy, New Jersey) of Monday 27^th April 1908:

When the breezy American lad turns upon the Irish alderman in “The Man of the Hour” ⁵ and sarcastically remarks “Now we’ll hear from Mr. Fixit,” there is always a roar of laughter. This line in itself is not so funny, but the way it is said and its peculiar fitness as applied to the diplomat who is trying to straighten out a youthful love quarrel never fails to land over the footlights.
“Mr. Fixit” is a bit of slang that appealed to the boys of Harvard and has become incorporated into the harmless persiflage in that famous institution of learning.
George Cohan’s slang classic “23” ⁶ was always somewhat obscure and is now seldom heard in polite society, but “Mr. Fixit,” as applied to the person who means well and makes a mess of it has a peculiar fitness that appeals to the young American sense of humor. The phrase was coined by an actor one day at rehearsal and so appealed to Mr. Broadhurst, the author, that he allowed him to keep it in.

⁵ The Man of the Hour is a play by the Anglo-American theatre owner-manager, director, producer and playwright George Howells Broadhurst (1866-1952); it opened at the Savoy Theatre, New York City, on Tuesday 4^th December 1906.
⁶ George Michael Cohan (1878-1942) was a U.S. entertainer, playwright, composer, lyricist, actor, singer, dancer and theatrical producer. The following is from Where Did “23” Originate, published in The Boston Daily Globe (Boston, Massachusetts) of Sunday 13^th May 1906:

Whence came “23,” that popular slang synonym for “get out,” “you have reached the end,” “quit,” etc?
[…]
All the disputants on the question of the birth of the expression agree that it was disseminated largely by a comedian named George Cohan, who used it in the musical comedy, “Little Johnny Jones.” But Mr. Cohan did not first employ the numeral as a slang phrase. He heard it and thought it a good line, so he used it.