"have the difficulty of" is correct and can be used in written English.
It is usually used to indicate that one is facing a challenge or some kind of hardship. For example, "The team had the difficulty of playing away from their home stadium."
Tory MPs campaigning in these seats have the difficulty of trying to win over voters at both ends of the spectrum: the Labour-Tory swing voters and the Ukip-Tory waverers.
Once again, we have the difficulty of interpreting a part of the formal semantics.
Although universities still have the difficulty of competing with attractive salaries in the nonacademic job market, Broeckx hopes that secondary facilities, such as day care for researchers with children, will convince young Ph.D.s to stay at the university.
"In a way, angels were like computers---very, very good at figuring things out and getting things done because they had rational minds but did not have the difficulty of having desires and passions, like humanity".
Her conversation often circles back to the choices women have, the difficulties of reconciling nurture and ambition, marriage and independence, what has been achieved and what has not.
But as demand has grown, so has the difficulty of finding and vetting qualified people.
But their reliability has increasingly come under scrutiny, as has the difficulty of doing recounts without a paper trail.
She then talked of her children, a bit about her ex-husband, the trouble they had had, the difficulty of being a single mother.
But the reliability of touch-screen machines has increasingly come under scrutiny, as has the difficulty of doing recounts without a paper trail.
The price had hardly changed since Van Twiller's day, and neither had the difficulty of figuring out how best to put the island to use.
Ordinary Iraqis, in the main, never had the difficulty of distinguishing fact from fiction that became a hallmark of their rulers.
